correlated electron transport in one-dimensional mesoscopic
TRANSCRIPT
CORRELATED ELECTRON TRANSPORT IN
ONE-DIMENSIONAL
MESOSCOPIC CONDUCTORS
a dissertation
submitted to the department of applied physics
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Na Young Kim
September 2006
c© Copyright by Na Young Kim 2006
All Rights Reserved
ii
I certify that I have read this dissertation and that, in
my opinion, it is fully adequate in scope and quality as a
dissertation for the degree of Doctor of Philosophy.
Professor Yoshihisa Yamamoto(Principal Adviser)
I certify that I have read this dissertation and that, in
my opinion, it is fully adequate in scope and quality as a
dissertation for the degree of Doctor of Philosophy.
Professor Malcolm Roy Beasley
I certify that I have read this dissertation and that, in
my opinion, it is fully adequate in scope and quality as a
dissertation for the degree of Doctor of Philosophy.
Professor David Goldhaber-Gordon
Approved for the University Committee on Graduate
Studies.
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Abstract
Mesoscopic systems have emerged as a result of advanced microfabrication processing.
These systems provide a new playground where tailor-made structures are available.
This enables the study of quantum phenomena due to dimensional confinement and
manipulation of system geometries. Two specific systems are investigated in this
dissertation: single-walled carbon nanotubes and quantum point contacts in a two-
dimensional electron gas.
Single-walled carbon nanotubes (SWNTs) provide a testbed to explore unique
quantum behaviors of one-dimensional (1D) systems. Unlike two- or three-dimensional
counterparts, in which the Coulomb screening justifies an independent single parti-
cle picture, the ground-state properties and system dynamics of 1D conductors are
deeply rooted in more complicated electron-electron interactions. One manifestation
of the 1D features is in electrical transport properties. Recently, the electrical con-
tacts between tubes and metal electrodes have been improved, allowing us to observe
quantum interference in ballistic SWNTs analogous to intensity fringes in Fabry-Perot
cavities. Electron transport properties of well-contacted SWNTs via measurements
of differential conductance and low-frequency shot noise are focused. Experimental
results exhibit strong correlations among conducting channels. The interpretation
of experimental observations within the Tomonaga-Luttinger liquid (TLL) theory is
discussed, which provides qualitative and quantitative agreements with experiments.
Especially, the characteristic TLL parameter inferred from the differential conduc-
tance and the current noise measurements agrees well with the theoretical values
predicted for SWNTs.
Quantum point contacts (QPCs) in a high-mobility two-dimensional electron gas
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(2DEG) system have been a prototypical device used to investigate low-dimensional
mesoscopic physics as well as a basic ingredient to explore quantum statistics of
particles. The quantized conductance manifests ballistic transport through QPCs and
it is well understood by the wave nature of quantum particles in terms of transmission
probability. An additional remarkable feature has been identified as the 0.7 structure,
reflecting many-body effects. An attempt to explore unresolved features in a QPC is
made with a control of the electron density in a 2DEG. Non-equilibrium transport
properties of differential conductance and current fluctuations in a backgated QPC is
characterized.
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Acknowledgements
“If I have seen so far
it is only because I have stood on the shoulder of giants.”
− Isaac Newton
Ph.D. stands for Doctor of Philosophy, but it has meant to me as a process to
learn Patience and Persistence, Humbleness and Humility through Devotion and
Dedication. Not to mention ruminate metaphysical questions over and over like who
I am, where my ways go, and where the truth resides. I admit that I have been
extremely blessed to walk this long journey with wonderful, supportive, precious
people together. These people have helped in various aspects make this thesis possible.
This is the moment I have been dreaming to express my deep gratitude in words
to cherished individuals from the bottom of my heart. Accumulating knowledge of
physics is an important asset I have had here at Stanford, furthermore I have been
incredibly lucky to build up the memorable relationship and friendship with people I
have encountered here.
First, I am deeply indebted to my research advisor, Professor Yoshihisa Ya-
mamoto. Recently, I realized that to do research is like to plant various seeds. Some-
times, you do know what the fruits or flowers of the seeds you plant are at the very
beginning. But sometimes, you plant them since they are seeds and see what the
fruits or flowers would be after gardening. From time to time, you thought you know
what the seeds are, but they turns out to be different than what you thought. Yoshi
has been a limitless source of research ideas and has been granting me freedom how to
garden seeds of research. His big picture and unique approach of physics have always
amazed me, and his tremendous patience and diligence have been exemplifying.
vii
My sincere thanks go to my reading committee, Professor Macolm R. Beasley and
Professor David Goldhaber-Gordon. Professor Beasley has impressed me, showing
humbleness as a mature image of scientists. I enjoyed his class about condensed
matter physics seminar series , which made the seminar approachable as well as made
me to grasp his systematic way to interpret physical concepts displayed in literatures.
Most of all, he is born-nature humorous so that I do now believe that physicists can be
funny and witty. Professor David Goldhaber-Gordon has been very supportive with a
great interest on my research, providing practical advices. His acute assessment and
sharp understanding always challenge me a lot. I learned the breadth and depth of
mesoscopic condensed matter field in his class and I will not forget the thrill of the
random draw to lead the mystery presentation each time.
I am grateful to have Professor Steve Kivelson who accepted the request to be
the chair of the Ph.D. oral committee without hesitation. In addition, his lectures
on superconductivity are one of the best lectures I have had at Stanford due to his
enthusiastic delivery and thorough knowledge in his field. I am very fortunate to
have Professor Katheryn Moler in my Ph.D. oral committe, who is always supportive
and encouraging. Her outgoing and open-minded personality have been unforgettable
since the first day I met her as a perspective student in 1999. One of the regret I
have had at Stanford is that I missed the opportunity to work for her as a rotation
student since I accepted the other offer one day earlier. As her teaching assistant, I
learned that how much she cares for students.
Many thanks are to the collaborators, without them anything in this thesis is
possible. Professor Hongjie Dai initiated the shot noise properties of single-walled
carbon nanotubes with great interest and passion. He has been very patient to accept
my slow pace of research progress. Most samples were prepared by Dr. Jing Kong who
seems to have no give-up and continuously to move forward. I miss a lot the moments
we started our days with prayers together. Other samples I could test were from Dr.
Jien Cao, Dr. Woong Kim and Dr. Ali Javey who are wonderful people whom I share
friendship. High-quality GaAs quantum point contacts have been provided by Dr.
Yoshiro Hirayama at the NTT basic laboratory, who also accepted me as a visiting
student in his lab to experience the fabrication processes and to interact with experts
viii
in his group.
I have been a true beneficiary of the rotation program in the department of applied
physics, who have had no real research experience during undergraduate. I am deeply
thankful for all professors who were willing to accept me as a rotation students,
involving in active research projects to taste what is the research and how to pursue it
: Professor Martin Greven who encouraged me to come one quarter early so that I got
accustomed to a new life at Stanford in advance and from whom I grasped the powder
study of high Tc superconductors with SQUID measurements; Professor Douglas
Osheroff who shared his graduate life to cheer me up when I was in trouble and
who welcomed me with a big smile whenever I bumped into him anywhere; Professor
James Harris who strongly recommended me to work on semiconductors over medical
physics to grasp the basics of material characteristics of GaAs; Professor Martin Fejer
who provided numerous ways and suggestions when I faced difficulties to quantify
material characteristics using proton exchange bath. In addition, I would love to
thank Professor Calvin Quate who always shows sincere and faithful enthusiasm in
our study, allowing me to use his atomic force microscopy in his lab and serving as
one of the qualifying examination committee. I also thank Professor Vahe Petrosian
and Professor Hari Manoharan for being the qualifying examination committee, who
endowed me to have a chance to swallow the fundamentals of physics.
I also take special time to express my gratitude to professors in my undergraduate
institute who have put faith in me that I can finish the degree: Professor Kwun, Sook-
Il who showed innocent passion on physics and loved me as his daughter; Professor
Kim, Sun Kee who was the first professor I can approach without any hesitation;
Professor Jhe, Wonho who prayed for me to fly with ambition and passion under
God’s will; Professor Char, Kookrin who advised me to enjoy and survive graduate
life whenever he visited Stanford over nice dinner; Professor Park, Yungwoo who
invited me as his summer students to have a chance to work on projects; Professor
Park, Byungwoo who supported me very much to receive the admission to Harvard
University with a fellowship which I ended up turned down and trusted me so much
that I could teach his daughter before I left abroad.
I am in debt to all members of Yamamoto group who have been always beside
ix
me whenever I need them. Especially, Dr. William D. Oliver is my first mentor who
has taught me everything with great patience, including all experimental engineering
techniques, theoretical models, and English. In addition he has advised me and
has supported me with his maturity. I cannot really express my deep gratitude in
words when he flew to Stanford for me on the Ph.D. oral defense in the mist of
his busy schedule. Dr. Recher is also my mentor from whom I learn the rigorous
theoretical approaches. Without him, the single-walled carbon nanotube project may
not be further explored and the experimental outcome may be in dark; however, he
persistently tries and manages to acquire quantitative and qualitative explanations.
His patience and persistence are what I would like to have in my personality. I am
grateful to Dr. Jungsang Kim who hosted me without hesitation when I requested the
lab tour of Stanford in 1996 even though I had never met him but since he is my senior
in the same undergraduate. He showed me the very room, Ginzton 25A to explain
his single photon devices very hard to me who had not taken Quantum Mechanics
class back then. I felt quite strange but comfortable when Ginzton 25A came to be
my destiny to commit my real Ph.D. life in 2000. I like to thank Dr. Xavier Maitre
and Dr. Robert Liu who were senior members of mesoscopic transport subgroup, who
influenced my research in various aspects. Professor Matsumoto and Professor Barry
Sanders who visited Y-group had listened to my circuit questions and had discussions
on the role of female physicists with introducing great books. Early generations of
Y-group nurtured me: Dr. Fumiko Yamaguchi, Dr. Matthew Pelton, Dr. Aykutlu
Dana, Dr. Charles Santori, Dr. Edo Waks, Jocylin Plant. Dr. Cyrus Master, Dr.
Thaddues Ladd, Dr. Hui Deng, Dr. Eleni Diamanti, Dr. Jonathan Goldman, Dr.
David Fattal have been friends as well as colleagues who make me feel lucky. Shinichi
has been a great office mate, teaching me the basic knowledge of optical setup and
helping me to overcome hurdles. Current members of y-group are just great as old
folks as my second family in the states: Dr. Bingyang Zhang, Dr. Kaoru Sanaka,
Katsuya Nozawa, Kai-mei Fu, Susan Clark, David Press, Y. C. Neil Na, Young Chul
Yun, Georgious Roumpos, and Kristiaan. In addition, Ms. Yurika Peterman, Ms.
Rieko Sasaki and Ms. Mayumi Hakkaku are the world-best administrators who really
take care of us as a family.
x
Spending two quarters in Fejer group, I have developed wonderful friendship with
Byer/Fejer group folks. Krishnan and Jonathan Kurz were great seniors I worked
with. Social outings with Loren, Frederick, Supriyo, Carlsten, David Hum and Yin-
wen have been joyful. I got to know Yin-Wen as a laser lab partner for two quarters
and thank her a lot for her innocent heart and affections. Moreover, we have had
great girly Christmas eve gathering and she always listens to my distress and concern,
standing in my side. Supriyo, Carlsten and David Hum are great people who make
me laugh and smile and who are always there for me when I need second hands in
the lab. My classmates, Tim, Jason, Adam, Brad, Yu-ju, Sylvia are in my heart and
memories. I am very lucky to have many great student advisors: Patrick Mang, Doug
King, Anu Tewary, Jonathan Goldman and Todd Sulchek, who really wanted me to
get well. Also, Seokchan, Jongkwan, Sanghyun, Yun and Harold have been extremely
helping me a lot to adjust here at the beginning in Korean styles. It was a real fun
to play basketball at 3 in the morning.
I am very grateful to many individuals for their wisdom and kindness with amaz-
ing technical support: Larry Randall for computer and machine shop techniques, Tom
Cover for clean room instruction, and Darla Le-Grand-Sawyer for smooth adminis-
tration.
Hana and Jungsuk have been my sister and brother, sources of encouragement,
who have cared me and my brother with love and prayers. NCBC church families
have been my another family in the states: Patty Kim, Linda Choi, Anthony Song,
Jane Lee, and Soohyun Cho.
I am grateful to my only brother, Sang Hoon who has inspired me with creative
new ideas and sharp questions, who has been my another hands in the lab whenever
I need while performing experiments, and who has challenged me to be professional.
I really do not know how much I depend on my lovely parents who have been patient,
crying with me when I am sad and laughing with me when I am happy and most
of all praying for me all the time at nights and at days. Their love and trust in me
push me to be at this moment and continue to push me to be further. My late grand
mother is still my source of strength and happiness, who prayed for me at 5 o’clock
every morning. Her prayers and love stand me up today. I DO love my family with
xi
my life.
Lastly and mostly, I cannot find the right words to thank our sincere Lord who
is always beside me, wakes me up every time I am in dark, heals me when I mess up
and stands me up when I am down. His soft and tender voice strengthen and guide
me. He has never been disappointed by me, loving me as who I am and teaching me
to be better. I would love to walk through the rest of my life with Him, to be the one
who He wants to be.
xii
Contents
Abstract v
Acknowledgements vii
1 Introduction 1
1.1 Mesoscopic Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Quantum Point Contact . . . . . . . . . . . . . . . . . . . . . 10
1.2 Electrical Transport Properties . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Classical Transport . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Many-Body Physics 18
2.1 Charge Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The Tomonaga-Luttinger Liquid Theory . . . . . . . . . . . . . . . . 28
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Mesoscopic Electron Transport 34
3.1 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Ballistic Transport . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Fundamentals of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Fluctuation and Dissipation Theorem . . . . . . . . . . . . . . 40
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3.2.2 General Formulation of Noise . . . . . . . . . . . . . . . . . . 42
3.2.3 Classification of Intrinsic Noise . . . . . . . . . . . . . . . . . 43
3.2.4 Crossover of Noise Sources in Frequency Domain . . . . . . . . 51
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Experiment Methodology 54
4.1 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Low-frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Single-Walled Carbon Nanotubes 71
5.1 Single-Walled Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . 72
5.1.1 Electronic Band Structure . . . . . . . . . . . . . . . . . . . . 73
5.1.2 Synthesis and Fabrication . . . . . . . . . . . . . . . . . . . . 86
5.2 Differential Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Quantum Interference . . . . . . . . . . . . . . . . . . . . . . 94
5.2.2 Spin-Charge Separation . . . . . . . . . . . . . . . . . . . . . 105
5.3 Low-Frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.2 Shot Noise and Fano factor versus the Drain-Source Voltage . 115
5.3.3 Fano Factor versus Transmission Probability . . . . . . . . . . 121
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Quantum Point Contact 126
6.1 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . . . . . 126
6.1.1 Energy Band Profile . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.2 Scattering Mechanism . . . . . . . . . . . . . . . . . . . . . . 131
6.1.3 Backgated 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Quantum Point Contact . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.1 Conductance Quantization . . . . . . . . . . . . . . . . . . . . 136
6.2.2 The 0.7 Structure . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3 Differential Conductance . . . . . . . . . . . . . . . . . . . . . . . . . 152
xiv
6.3.1 Non-integer Conductance Plateaus at Finite Bias Voltage . . . 155
6.3.2 Density Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4 Low-frequency Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . 158
6.4.1 Noise Suppression at Non-integer Conductance Plateaus . . . 160
6.4.2 Density Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7 Conclusions 165
A Physical Constants 167
B Conversion Tables 168
B.1 Energy and temperature . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.2 Frequency, temperature, energy, wavelength and time . . . . . . . . . 169
C Statistics of Particles 170
C.1 The Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . . . 170
C.2 The Fermi-Dirac Distribution . . . . . . . . . . . . . . . . . . . . . . 171
C.3 The Bose-Einstein Distribution . . . . . . . . . . . . . . . . . . . . . 174
C.4 Basic Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 174
D The Dirac Delta Function 176
D.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
D.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
E Useful Mathematical Formulas 178
E.1 Even and Odd functions . . . . . . . . . . . . . . . . . . . . . . . . . 178
E.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
E.3 Fourier-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
E.4 Pauli Spin Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
E.5 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 180
E.6 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
E.7 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
xv
F Recipe of Making Printed Circuit Boards 184
Bibliography 187
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List of Tables
1.1 Classified dimension of mesoscopic conductors systems. . . . . . . . . 5
1.2 Classified quantum electron transport . . . . . . . . . . . . . . . . . 15
5.1 The g values from the power-law scaling analysis of four samples. . . 121
5.2 The relation between extreme values of F and the overall transmission
probability T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xvii
List of Figures
1.1 Zone of macroscopic, mesoscopic and microscopic systems. . . . . . . 3
1.2 (a) Semiclassical cartoon of 6C atomic shell structure. (b) The diagram
of s - and p - orbital hybridization: tetragonal sp3 (left) and planer
sp2 (right). (c) Structures of carbon based material: zero-dimensional
C60 (left), three-dimensional diamond (middle) and three-dimensional
graphite (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 History of carbon nanotube research from the discovery to the present
research area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 (a) The current and voltage measurements of a 23.5-kΩ (manufac-
turer’s rating) surface mount chip resistor with HP 4145B semiconduc-
tor parameter analyzer. A linear regression analysis gives conductance
G ∼ 41.316 µS or R ∼ 24.204 kΩ. (b) Measured current in time at 5
mV bias to the surface mount chip resistor. . . . . . . . . . . . . . . 13
2.1 (a) The Fermi sphere of non-interacting electron systems. (b) The
momentum distribution function of the Fermi gas systems. (c) The
spectrum of particle-hole excitations in the Fermi gas systems. . . . 25
2.2 The momentum distribution of the Fermi liquid system. . . . . . . . 27
2.3 The TLL parameter g as a function of Rs/R. Two red lines are marked
at g = 0.2 and g = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 (a) An one-dimensional ballistic conductor in a two-terminal configu-
ration. (b) The energy dispersion of free electrons in a reservoir (left)
and a conductor (right). . . . . . . . . . . . . . . . . . . . . . . . . . 36
xviii
3.2 A two-port system represented by second quantized operators. . . . 38
3.3 Spectrum of dominant noise sources in frequency domain. . . . . . . 43
3.4 A parallel resistor-inductor circuit. . . . . . . . . . . . . . . . . . . . 44
4.1 (a) Two-terminal and (b) four-terminal measurement schematics with
a constant current bias. . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 (a) Johnson-Nyquist noise vs. temperature for different resistance.
(b) Johnson-Nyquist noise of resistors at various temperatures: room
temperature (293 K), liquid nitrogen temperature (77 K), liquid helium
temperature (4 K) and 1 K. . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 (a) The 4 K home-made dipper and (b) Oxford Helium 3 sorption
cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 (a) Estimated dc (dot) and ac (square) voltages at a 1 nA current bias
to a variable resistor together with Johnson-Nyquist noise measured by
a 100 kHz equipment at room temperature and 4 K. Assume ac signal
is hundred times smaller than the dc value. (b) Johnson-Nyquist noise
with a 1 Hz bandwidth at 4 K. . . . . . . . . . . . . . . . . . . . . . 62
4.5 (a) Shot Noise and Thermal Noise crossover. (b) The threshhold R
and I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 (a) The circuit diagram of AC modulation scheme. (b) The square-
wave voltage in time. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 (a) SONY and (b) Fujitsu FSU01LG MESFET bias response at room
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 The photograph of a cryogenic amplifier and resonant tank circuit used
in the single-walled carbon nanotube show noise experiments. . . . . 70
5.1 (a) Graphite lattice structure. (b) The direct and (c) the reciprocal lat-
tice space of graphene with unit vectors ~ai, ~bi and translational vectors
~ri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Energy band structure of Graphene. . . . . . . . . . . . . . . . . . . 78
5.3 SWNT geometry on the graphene lattice structure. Chiral vector ~Ch
is drawn for a specific case, a (4,2)-SWNT. . . . . . . . . . . . . . . 79
xix
5.4 Armchair SWNT real (a) and reciprocal (b) lattice space. (c) (10,10)
energy band structure. The first BZ of SWNT is indicated by two thick
vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5 (a) The µ = 0 band for a (10,10) SWNT along symmetry points Γ, K,
and K ′. The π, π∗ wavefunctions are clearly denoted based on the
energy band coefficients. The real and imaginary coefficients of upper
energy band (b) and lower energy band (c) for µ = 0. The dotted line
is the zone boundary of graphene. . . . . . . . . . . . . . . . . . . . 83
5.6 A zigzag SWNT unit cell in a real (a) and a reciprocal (b) lattice space.
(c) Semiconducting zigzag (10,0) energy band structure. The first BZ
of SWNT is indicated by two thick vertical lines. (d) µ = 6 band for
(10,0). The real and imaginary coefficients of upper energy band (e)
and lower energy band (f) for µ = 6. . . . . . . . . . . . . . . . . . . 84
5.7 (a) The µ = 6 band for a (10,0) zigzag SWNT. The real and imaginary
coefficients of upper energy band (b) and lower energy band (c) for the
µ = 6 band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.8 (a) Schematics of CVD chamber. (b) The mechanism of SWNT growth
from catalysts in CVD chamber adapted from K. J. Cho group. . . . 88
5.9 The schematics of SWNT-device fabrication processes. . . . . . . . . 90
5.10 (a) Optical microscopy picture of portion of a chip containing wire-
bonded devices, (b) a zoom-in view of an individual device, and (c)
atomic force microscopy image of an individual SWNT with patterned
Ti/Au electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.11 (a) The schematics of three-terminal SWNT device. (b) Experimental
two-dimensional image plot of differential conductance versus a drain-
source voltage in y-axis and a backgate voltage in x-axis. . . . . . . . 95
5.12 The differential conductance from a 360 nm-long SWNT device at Vg
= - 5 V. Experimental data are blue circles and the theoretical fitting
of the single-channel double-barrier structure model is in red. . . . . . 97
xx
5.13 (a) Diagram of two-channel double barrier system. (b) Two-dimensional
image plot of differential conductance versus drain-source voltage in y-
axis and backgate voltage in x-axis. . . . . . . . . . . . . . . . . . . . 98
5.14 Illustration of the TLL model on a SWNT device. . . . . . . . . . . 101
5.15 (a) The Vds-dependent f1(Vds, g, T ) at T = 4 K for g = 0.25 (red),
g = 0.75 (green) and g = 1 (blue). (b) The Vds-dependent f2(Vds, g, T )
at T = 4 K for g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). . . 104
5.16 Differential conductance versus Vds at a certain Vg. (a) Experiment
and (b) Theory based on TLL for g = 0.25 (red) and g = 1 (blue). . . 106
5.17 Differential conductance versus Vds at different backgate voltages for
experiment results (a), the theoretical plots from non-interacting Fermi-
liquid theory (b) and from the Tomonaga-Luttinger liquid theory (c). 107
5.18 Schematics of SWNT shot noise measurement setup. . . . . . . . . . 110
5.19 The 1/f noise crossover of SWNT devices for two temperatures 293 K
and 77 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.20 The coupling efficiency α between ILED and IPD at (a) T = 293 K and
(b) T = 4 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.21 Full Shot Noise from the LED/PD pair (a) T = 293 K and (b) T = 4 K 114
5.22 A representative log-log plot of low-frequency shot noise from the
LED/PD pair (upside-down triangle) and the SWNT (diamond) as
a function of Vds. The straight line is the outcome of linear regression
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.23 (a) The integration in Vds of the Vds-dependent f1(Vds, g, T ) at T = 4
K for g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). (b) The
integration in Vds of the Vds-dependent f2(Vds, g, T ) at T = 4 K for
g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). . . . . . . . . . . . 118
5.24 The experiment data (blue square) of the SWNT noise with the theo-
retical fitting plot(red). . . . . . . . . . . . . . . . . . . . . . . . . . 119
xxi
5.25 A representative log-log plot of Fano factor (open diamond) obtained
from experiments against Vds together with theoretical theoretical fit-
ting of Tomonaga-Luttinger liquid theory for g = 1 (straight line) and
g = 0.25 (dotted line). The broken line on the experimental data
represents the power-law scaling analysis. . . . . . . . . . . . . . . . 120
5.26 Fano factor versus transmission probability taken at Vds = 40 mV
from five SWNT-devices (filled symbols) at varying Vg values. (a)
Ballistic phase-coherent transport theory for one-(dark blue straight)
and two-channel (dotted area) models. (b) Phase-incoherent picture
theory for distributed elastic (square) and inelastic (diamond), inco-
herent double-barrier (light green straight) and many-barrier model
(dark green straight) . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.1 (a) Band profile of a GaAs/AlGaAs heterostructure. (b) Band profile
of a 2DEG embedded in a doped GaAs/AlGaAs heterostructure. . . 128
6.2 (a) Growth structure of backgated 2DEG. d1 and d2 are the thickness
of GaAs and the two layers of AlGaAs and superlattice barriers respec-
tively. (b) Band diagram of backgated 2DEG operation. A thick solid
line represents the case of Vb,th, a dotted line is for the above Vb,th and
a thin solid line is for the below Vbg,th. . . . . . . . . . . . . . . . . . 134
6.3 (a) Measured Hall voltage as a function of external magnetic field per-
pendicular to 2DEG by changing the backgate voltages VBG. (b) The
calculated electron density of 2DEG versus VBG. . . . . . . . . . . . 135
6.4 (a) Schottky-split techniques to form a QPC in a 2DEG sysgem. (b)
Sketch of energy dispersion of electron reservoirs and QPC . . . . . . 137
6.5 (a) The saddle-point potential for |ωy/ωx| = 2. (b) computed differen-
tial conductance for the saddle-point potential (a). . . . . . . . . . . 140
6.6 (a) The actual voltage drop effect in left and right moving channels.
(b) Computed differential conductance for non-zero Vds cases. . . . . 141
6.7 The 0.7 structure from differential conductance measurements at 1.5 K. 143
xxii
6.8 (a) The band structure without spin-orbit interaction under zero mag-
netic field ~B = 0 for the first five n. (b) The band structure with
non-zero spin-orbit interaction at ~B = 0. (c) The band structure with
non-zero spin-orbit interaction at finite magnetic field. In all three
cases magnetic field is perfectly aligned in x-direction, i.e. θ = 0. P
and ξ are defined in the context. . . . . . . . . . . . . . . . . . . . . 148
6.9 Computed differential conductance as spin-orbit coupling in a simple
harmonic oscillator potential is on while the magnetic field is kept zero
in (a) and the magnetic field is 0.05 in the unit of g∗µBB/2~ω in (b). 150
6.10 Computed differential conductance as spin-orbit coupling is on while
magnetic field is kept zero. . . . . . . . . . . . . . . . . . . . . . . . 151
6.11 (a) Scanning electron microscope (SEM) image of a quantum point
contact in a AlGaAs/GaAs 2DEG. (b) The Hall-bar structure of the
wirebonded device taken by SEM. . . . . . . . . . . . . . . . . . . . 153
6.12 (a) Experimental differential conductance dG by a sweep of Vg at fixed
Vbg = 2.3 V at finite Vds (b) Transconductance dG/dVg (c)Vds depen-
dence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.13 (a) Observed differential conductance traces at each Schottky gate volt-
age as a function of Vds. All date were taken at Vbg = 2.3V and 1.5 K.
(b) Computed differential conductance with the Vds-dependent saddle-
point potential up to the linear term, i.e. γ = 0. Including second-order
corrections in Vds with two opposite signs of the coefficient, (c) γ > 0
and (d) γ < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.14 The tuning variables are Vg and Vbg and dG is measured at four-probe
techniques with ac signal Vac ∼ 50 µV and the dc bias (a) Vds = 0 mV
and (b) Vds = 2 mV. The Vbg varies from 3.0 V (rightmost) to 2.58 V
(leftmost) by 0.01V interval. . . . . . . . . . . . . . . . . . . . . . . 157
6.15 Bias dependence at Vbg = 2.3 V. (a) Vds = 0.7 mV. (b) Vds = 2 mV.
(c) Vds = 2.5 mV. (d) Vds dependent conductance. . . . . . . . . . . 159
xxiii
6.16 (a) Two-dimensional plot of shot noise raw data at Vbg = 2.3 V with
lines at the conductance values. (b) The contour plot of conductance
values in the unit of GQ simultaneously taken with shot noise measure-
ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.17 Two-dimensional image plot of shot noise versus Vg and Vbg at (a)
Vds = 1 mV. (b) Vds = 2 mV. The straight lines with number are
corresponding to the conductance values normalized by GQ. . . . . . 163
xxiv
Chapter 1
Introduction
Chance favors the prepared mind.
− Louis Pasteur
“Meso-” means middle in Greek, µǫσoς. The term, “mesoscopic system”, was first
coined in 1976 by N. G. van Kampen in the context of statistical physics [1], and
similar usages have come to appear more often in diverse fields since early 1990s
with the advent of microfabrication processing techniques, referring to one whose
dimension lies in between microscopic and macroscopic counterparts. This middle one
has attracted more attention, providing ample room to investigate physical inquiries.
Physicists have been seeking the origin of nature on earth even in a far-off galaxy
with human beings’ imagination and empirical justification. The scope of physics is
very broad, ranging from bulk systems (macroscopic world) to exotic materials re-
sulting from diverse electron configuration (microscopic world). In modern days, the
subject of physics becomes rather subdivided into diverse areas according to primary
interests: for example, astrophysics for universe, particle physics for ultimate con-
stituent objects in atoms, biophysics for biological entities, condensed matter physics
for solid systems and more. Condensed matter physics is particularly concerned
about states and phases of not only naturally existent but also engineered material,
searching for the microscopic level understanding.
1
2 CHAPTER 1. INTRODUCTION
The naturally accessible systems are found in both macroscopic and microscopic
level, but the governing principles of phenomena occurring in two realms are quite
distinct. The statics and kinetics in macroscopic world has been completely assessed
by the deterministic picture in classical Newtonian physics. For example, given an
initial position of a particle with a velocity at a certain time, any succeeding states
at later times can be readily computed along a distinguishable particle trajectory.
This picture, however, turned out to be an inadequate eye to examine properties and
behaviors of individual and aggregates of constituents at the atomic and subatomic
levels. To perceive correct understanding of phenomena in such environment is beyond
the realm of classical physics, requiring alternative paradigm to replace deterministic
reduction of systems.
A tremendous insight has been gained in early twentieth century by brilliant sci-
entists who successfully and beautifully established theoretical framework quantum
physics, game of chance or probability. Quantum physics describes statics and kinet-
ics in microscopic world by introducing the description of wavefunction as a solution
to Schrodinger equation to incorporate the intrinsic particle-like and wave-like nature
of quantum entities. Furthermore, quantum world exhibits substantial character-
istics: duality of wave and particle, indistinguishability, quantization, Heisenberg
uncertainty, finite zero-point energy, and quantum entanglement. The duality fea-
tures coherence in the dynamics of quantum particles in quantum world, linking to
correlation effects in systems. Quantum particles in the coherent sate have a well-
defined energy and a well-defined phase configuration. In the beginning of founding
quantum theory, only Gedanken-experiments with coherent particles had been pos-
sible for validate postulates and predictions of quantum theory due to hindrance to
approach microscopic world. As closing the gap, the new playground to perform
previous gedanken experiments in the lab, mesoscopic world has recently emerged.
What enables us to enlarge our focus beyond preformed area is engineering improve-
ment of fabrication technology and material synthesis. Sub-millimeter structures are
repeatedly patterned by a photolithography, and even smaller features down to tens
of nanometers are constructed by electron-beam lithography with a skillful care in
a reproducible rate. The combination of two lithography methods yields simple and
3
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Figure 1.1: Zone of macroscopic, mesoscopic and microscopic systems.
4 CHAPTER 1. INTRODUCTION
complex device structures. Of numerous methods in the latter synthesis area to
produce new material, molecular beam expitaxy and chemical vapor deposition are
widely used at present. One outcome of such efforts are mesoscopic conductors.
1.1 Mesoscopic Conductors
Mesoscopic conductors have exhibited subtle and sophisticated phenomena which
are deeply rooted in quantum mechanics and are tightly associated with many-body
interactions. An easy means to classify macro-, micro- and meso-scopic conductors
is physical length. Figure 1.1 presents three regions of systems along the length scale
bar with familiar objects corresponding the size of physical length. Macroscopic ones
are often what our bare eyes can see and what our hands can fold, typically bigger
than millimeters, whereas microscopic systems are too tiny to visualize how behaviors
of atoms are without supplementary equipments, roughly around 0.5 - 10 A close to
the atomic sizes, where A is 1 over 10 billion of millimeter. Mesoscopic world closes
the gap between two regions, spanning from millimeters to nanometers (a billionth of
a millimeter).
The manipulation of system size enables us to obtain all dimensional mesoscopic
conductors. The classification of system dimension is achieved by the comparison
between the physical system length and Fermi wavelength. Fermi wavelength λF is
defined as a wavelength of carriers at the Fermi level, λF = h/pF = 2π/kF where h is
the Planck’s constant and pF, kF are the momentum and wavenumber at the Fermi
surface. Since conduction in mesoscopic systems is dominated by electrons at the
Fermi level, λF means the wavelength of major carriers. λF in metals is typically
atomic dimension, around A, whereas λF in semiconductors is about 10 - 100 nm.
1.1. MESOSCOPIC CONDUCTORS 5
Suppose Lx, Ly and Lz represent physical length of system, satisfying Lx < Ly <
Lz without any loss of generality. Table 1.1 exhibits the condition of each dimension.
Dimension ConditionOne Lx, Ly < λF < Lz
Two λF ∼ Lx ≪ Ly, Lz
Three λF ≪ Lx, Ly, Lz
Table 1.1: Classified dimension of mesoscopic conductors systems.
Two one-dimensional (1D) mesoscopic conductors are studied in this dissertation:
carbon nanotubes and quantum point contact in GaAs/AlGaAs semiconductor het-
erostructures.
1.1.1 Carbon Nanotubes
Carbon (6C), the sixth element in the periodic table, is one of the simple and familiar
atoms. It ubiquitously appears either homogeneous or heterogeneous compounds in
diverse forms from in the atmosphere, on earth, in the sea to inside living bodies.
The abundance of carbon-based chemicals results from three hybridization arrange-
ments of four valence electrons in 2s and 2p orbitals: sp, sp2, and sp3. Even with
carbon element only, different materials are found and any dimensional structures
exist: graphite and diamond in three dimension, graphene in two dimension and the
bulkyball fullernene C60 in zero shown in Fig. 1.2. Carbon bondings are mysterious
that they can be as hard as diamond to scribe other crystals or they can be as weak
as graphite to scribble on paper in a certain circumstance.
In 1991, carbon strands in a furnace for the fullerene production were fortuitously
observed [2]. Since strands which looked like concentric cylinders were unprecedent,
they were denoted as ‘microtubules’ [2], ‘fullerene tubules’ [3] or ‘graphene tubules’ [4]
at first in literatures. These titles reflected different perspectives as to how they were
formed, cylindrical tubes are either elongated fullerene mutations or roll-up wires
with several layers of graphene. Both views envisioned them as 1D objects regardless
6 CHAPTER 1. INTRODUCTION
(c)
(b)
(a)
(c)
(b)
(a)
Figure 1.2: (a) Semiclassical cartoon of 6C atomic shell structure. (b) The diagramof s - and p - orbital hybridization: tetragonal sp3 (left) and planer sp2 (right). (c)Structures of carbon based material: zero-dimensional C60 (left), three-dimensionaldiamond (middle) and three-dimensional graphite (right).
1.1. MESOSCOPIC CONDUCTORS 7
of whether they start from zero or two-dimensional mother material. Geometric ratio
of diameters and lengths supports the 1D point of view since diameters are around
tens of nm and lengths are around microns even mm. Possibility to access 1D systems
from tiny carbon tubules with ease was of great interest both to scientists who have
longed for a medium to explore unique 1D properties and to engineers who have
searched alternative material to attain further miniaturization of electronic circuits.
Condensed matter theorists immediately approached and investigated them af-
ter the discover of carbon nanotubes, taking the second view of tubules from two-
dimensional graphene. They simplified and attacked a model problem, one single
tube from one graphene layer in 1992 [3, 4]. Attempts to compute electronic band
structures were readily pursued by applying additional boundary condition resulting
from a spatial confinement to the well-established graphite band structures [5]. The
band structure calculation predicted a surprising insight that the tubules would be
either metallic or semiconducting according to the size of tubes and a roll-up di-
rection [3, 4]. These insightful theoretical prediction launched active research field
with carbon nanotubes (CNTs) since early 1990s. Many terminologies were coined
including single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs)
in order to identify one or more than one tubes, armchair and zigzag tubes inspired by
shapes of carbon hexagon on surfaces. Another landmark in the history of CNT re-
search area emerged in the year 1993, the discovery of theoretically imagined SWNTs.
Real material to test theoretical investigation became alive. This event blossomed the
field furthermore to embrace more and more communities from physicists, material
scientists, chemists to electrical, mechanical, chemical engineers.
The first phase of carbon nanotube experiments was rather limited into material
level investigation such synthesis [2,6–8], chemical treatment [9–11], and surface imag-
ing by scanning electron microscopy [2,8], atomic force microscopy and scanning tun-
nelling microscopy [12,13]. The upturn to obtain transport properties were achieved
once experimentalists found a way to couple grown-nanotubes to metal electrods in
late 1990s [?,14]. In the electron transport context, two tasks were targeted for better
understandings: one was to synthesize nanotubes at designated place and the other
was to improve the coupling between the tube and the electrodes. The first task
8 CHAPTER 1. INTRODUCTION
Discovery of MWNT
Material characterization
First electrical transport: Coulomb blockade
2002
1991
1993
1992
1998
1997
1999
2000
2001
Discovery of SWNT
Chemical Vapor Deposition SWNT growth
Optical luminescence and florescence
Room temperature single-electron transistor
Orbital Kondo effect
Integrated SWNT device on Si-wafers
Ballistic SWNT devices
Tomonaga-Luttinger behavior in tunnelling regime
Kondo effect
Bandstructure calculation of CNT
2005
2004Ahranove-Bohm effect
Figure 1.3: History of carbon nanotube research from the discovery to the presentresearch area.
1.1. MESOSCOPIC CONDUCTORS 9
was overcome by the chemical vapor deposition technique to produce high-yield and
high quality of SWNTs nearby catalyst islands [15], and the second task was continu-
ously attempted with different metal electrodes and annealing, making nearly Ohmic
contact recently [16–18]. Furthermore, the investigation of unique electronic proper-
ties was expedited as integrated nanotube circuits was accomplished [19]. Numerous
quantum transport phenomena induced from phase-coherent electrons and strong in-
teractions among electrons have been continuously revealed in SWNTs coupled to
electron reservoirs summarized in Fig. 1.3: Coulomb blockade [20–22], Tomonaga-
Luttinger liquid behavior [23,24], quantum ballistic interference [16,17], Kondo [25,26]
and orbital Kondo [27], Aharonov-Bohm interference [28] and magnetic orbital mo-
ment determination [29]. Ballistic transport here means electron conduction does not
experience any types of scattering and the detailed discussion on transport regime is
given in the following subsection.
Along attaining fundamental physical knowledge, advancement of nanotube tran-
sistors has been achieved by making field effect transistors with semiconducting
tubes [18, 20], and interplay of electrons and phonons in transistor-devices was ex-
amined [30–32]. In addition, CNTs showed superior sensitivity to chemical sensor
applications [33]. The extensive study of electrical transport properties leads to ex-
ploring novel quantum Hall effect in graphene very recently [34,35]. Recent interests
span to optical properties based on electronic band structures of semiconducting
carbon nanotubes from exciting florescence spectroscopy [36] and light emission in
field-effect transistor structures [37]. Further photoluminescence [38, 39] and elec-
troluminescence [40–43] results are in question, studying strong exciton effects in
one-dimensional system and illuminating possibilities of optoelectronic components
in future. Growing interests in CNTs since the beginning of discovery is based on
aforementioned novel properties and fabrication advantages of low cost and defect-free
crystalline structures. A big challenge in this mature field is to synthesize particular
types of nanotubes in a controllable manner.
10 CHAPTER 1. INTRODUCTION
1.1.2 Quantum Point Contact
There are not so many semiconductor elements in nature, and they are neither good
conductors and nor good insulators. But what makes semiconductor materials preva-
lent in everyday life is that they can be tuned and manipulated with one parameter,
their inherent energy gap between conduction and valence bands. They are insulat-
ing but become conducting by low energy excitations. Its adjustability has greatly
revolutionized contemporary life patterns in terms of computational power, telecom-
munication, and electronic gadgets under the motto, faster, smaller, and quieter. Out
of amazing technology breakthroughs, a band engineering drives new material using
various combinations of different semiconductor elements and compounds. Semicon-
ductor Heterostructures composed of more than one element is one of such outcome
by the band engineering.
Semiconductor heterostructures are in general made by epitaxy methods, whose
advantages are high-purity crystalline layer production, thickness control, and easy
material switching. Epitaxy methods are further differentiated according to phase
of sources into liquid phase epitaxy, vapor phase epitaxy and molecular beam epi-
taxy [44]. Fabrication methods have endowed the semiconductor field to control fun-
damental parameters like bandgap energy, effective mass of carriers and mobilities.
There is no fundamental hindrance to mix any combination of semiconductor ele-
ments to form heterostructures; however, there is practical reason not to produce any
random combination of materials. Of importance is to select materials whose lattice
constants are close to each other since strain and roughness are induced at interfaces
of two different materials, encountering strain and roughness scattering. The quality
of interfaces directly affects mobilities of carriers, thus it would be a crucial factor to
shape heterostructures. At present, Group III-V semiconductor heterostructures are
a central system of physics and engineering research.
Semiconductor heterostructures are diversified as two-dimensional quantum well,
two dimensional electron gas systems, one dimensional quantum wires and zero di-
mensional quantum dots. Quantum point contact (QPC) is one of the simplest device
structures on top of two-dimensional electron gas systems. Early interests on QPC
1.2. ELECTRICAL TRANSPORT PROPERTIES 11
were closely linked to the empirical test of quantum conductance behavior in a bal-
listic regime based on theories in 1960s [45]. Unlike metallic point contact formed
in a crude way by adjoining two sharply-edged metals [46], QPC in semiconductor
are fabricated in complicated but reproducible steps of microfabrication processing.
A split-gate technique are a common method to make QPCs based on the idea that
adjusting negative voltage to electrodes on top of the two-dimensional gas systems
induces controllabe electrostatic potential profile underneath. Indeed, nice quantized
plateaus were first observed with a good agreement with theory as a representation
of the ballistic transport in GaAs/AlGaAs heterostructures [47, 48]. A single QPC,
however, still has unresolved feature in conductance traces, so-called “0.7 structure”,
which is on an active research target at present.
QPCs are known as quasi-one dimensional waveguides of degenerate electrons at
low temperatures and sources of monochromatic electrons. Thus, they are elemen-
tary components to realize electro-optics [45] including focusing capability of coherent
electrons under magnetic field. In future, a single and series of QPCs will be expected
to form a solid state qubit for quantum information processing and quantum compu-
tations [49].
1.2 Electrical Transport Properties
Characterizing materials and matter in nature is driven by a yearn for underlying
physical principles which govern nature’s fundamental entities. The principles may be
revealed in ground state properties, excitation spectra and order parameters in phase
transitions. There exist various probes to extract such information which is expected
to be converged to simple governing principles in nature via scattering processes
caused by photons, x-rays and neutrons or transport processes.
Transport properties are obtained from observing a response to an external stim-
ulation according to material characteristics. Thermal transport probes heat con-
duction and electrical transport does charge conduction across materials. In most
cases, the relation between the stimulus and the response is assumed to be linear and
the ratio of the two quantities does carry system information. Similar to conductor
12 CHAPTER 1. INTRODUCTION
divisions in the previous section, various transport regimes are identified. In order
to appreciate quantum transport as a theme of this dissertation, the discussion of
transport begins with classical transport.
1.2.1 Classical Transport
According to the trend of electrical conductivity with respect to temperature, bulk
elements and compounds are classified into metals or insulators. Metallic material
displays the linear response between current and voltage, so-called Ohm’s law. The
famous Ohm’s law is a good measure of classical electron transport, in which the coeffi-
cient of linear relation between measured current(voltage) and biased voltage(current)
is called resistance. Resistance depends on intrinsic material conductivity(resistivity)
and the geometry of conductors. The origin of resistance in bulk systems is explained
by scattering processes. Electrons or charge carriers are greatly scattered by defects,
impurities or other carriers in materials. The overall scattering processes resulted
from diverse sources become obstacles, reducing conductivity.
Resistance is a macroscopic statistical observable since many electrons are involved
in electrical conduction process. More accurately in this statistical point of view,
resistance is the average value or mean of the ratio of measured current and biased
voltage. There are fluctuations about mean values in the time domain, often called
as noise in electronic circuits. Figure 1.4 presents an example of electrical transport
measurements with a macroscopic resistor. As expected, the linear relation of current
and voltage is shown in Fig. 1.4 (a). The time-trace of current through the same
resistor at a fixed bias voltage V = 5 mV is displayed in Fig. 1.4 (b), exhibiting time
fluctuations. This fluctuation or noise corresponds to variance about the mean in
statistics. The significance of noise lies in two-fold: (1) noise limits the accuracy of
measurement outcomes and (2) intrinsic noise regarding dynamics of charge carriers
delivers transport information.
1.2. ELECTRICAL TRANSPORT PROPERTIES 13
-200x10-9
-100
0
100
200
I (A
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204.4x10-9
204.2
204.0
203.8
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I (
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(b)
-200x10-9
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100
200
I (A
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100
200
I (A
)
-4x10-3
-2 0 2 4
V (V)
Gavg = 41.316 µS
(a)
204.4x10-9
204.2
204.0
203.8
203.6
I (
A)
20151050
Time (s)
Iavg = 203.995 nA
σ (std) = 9.041*10-11
(b)
204.4x10-9
204.2
204.0
203.8
203.6
I (
A)
20151050
Time (s)
Iavg = 203.995 nA
σ (std) = 9.041*10-11
(b)
Figure 1.4: (a) The current and voltage measurements of a 23.5-kΩ (manufacturer’srating) surface mount chip resistor with HP 4145B semiconductor parameter analyzer.A linear regression analysis gives conductance G ∼ 41.316 µS or R ∼ 24.204 kΩ. (b)Measured current in time at 5 mV bias to the surface mount chip resistor.
14 CHAPTER 1. INTRODUCTION
1.2.2 Quantum Transport
Discerned from classical transport, features of quantum transport are originated from
aforementioned unique traits such as coherence and quantization well described by
quantum physics. Quantum transport is further resolved into several regimes mainly
by lengthscale comparison. These transport divisions are crucial to understand meso-
scopic conductor transport properties. Besides Fermi wavelength, other relevant char-
acteristic lengths are defined for classified transport regimes: (1) mean free path, lmfp,
(2) thermal diffusion length, lT, and (3) phase coherence length, lφ.
Mean Free Path lmfp
Mean free path, as the name indicates, is an average distance in which particles can
move freely. The hindrance to free motion is due to scattering by defects, impurities
or grain boundaries. Elastic scattering does not conserve momentum but energy,
while inelastic scattering changes both momentum and energy of incident particles.
Thus, mean free paths due to elastic and inelastic scattering should be differentiated
although generally lmfp refers to the elastic mean free path. In semiconductors lmfp
is closely related to the mobility of carriers, and in metals lmfp is much longer than
λF. As lmfp becomes comparable to λF, systems with such lmfp are called in the dirty
limit.
Thermal Diffusion Length lT
At non-zero temperatures, electron wavepackets have energy width about kBT where
kB is the Boltzmann’s constant and T is the temperature. This energy uncertainty
induces diffusion in time. lT is a characteristic length of diffusion process due to
thermal energy.
Phase Coherence Length lφ
Within lφ, particles preserve their phase. Dynamical interactions including mutual
Coulomb interactions among electrons and electron-phonon interactions disturb phase
1.2. ELECTRICAL TRANSPORT PROPERTIES 15
Regime ConditionBallistic Lx, Ly, Lz < lmfp, lT, lφDiffusive lmfp, lT ≪ Lx, Ly, Lz
Dissipative lφ < Lx, Ly, Lz
Table 1.2: Classified quantum electron transport
coherence. Therefore, this length is important to determine whether quantum inter-
ference effects from phase coherent sources can be detectable or not in systems.
Comparisons of such scales define three distinct transport regimes in Table 1.2.
Varying the physical length of mesoscopic conductors, all enlisted transport regimes
are indeed within practical reach. Condition states that both in dissipative and in
diffusive regime, transport quantities are dominated by scattering process similar to
classical case. In detail, dissipative conductors suffer from inelastic as well as elastic
scattering losing previous information of momentum and energy, whereas diffusive
conductors have elastic scatterers, preserving momentum but not energy. For the
ballistic regime, on the other hand, all dimensions of ballistic conductors are much
smaller than all length scales, namely electrons participating in conduction process
do not encounter any kinds of scattering sources without modifying momentum and
energy.
As an extension from classical argument between resistance and scattering, bal-
listic conductors are not resistive at all in principle. It is true and indeed confirmed
empirically with a special care to eliminate the contact resistance between electron
reservoirs and a ballistic conductor [50]. It implies that ballistic conductors in mea-
surements have non-zero resistance, but it comes not from scattering processes but
from electron modes selected at the interface of a reservoir and a conductor. There
needs to be an alternative way to express resistance beyond Ohm’s law. Landauer
captured the significance of the wave nature of charge carriers in mesoscopic conduc-
tors, and he developed the theory to estimate resistance or conductance in terms of
transmission probabilities of propagating electron modes analogous to electromagnetic
16 CHAPTER 1. INTRODUCTION
photon modes. He predicted a finite resistance of mesoscopic conductors connected
to electron reservoirs at both ends without introducing scattering [51]. His prediction
which was back then at the heart of controversy against the classical perspective of
resistance had driven intensive experimental efforts on ballistic transport by design-
ing appropriate device structures and geometries QPC in previous section closed the
controversy with observation of conductance plateaus in the ballistic regime [47,48].
Previous perspectives to envision resistance properties are based on single and
independent particle picture. The final quantity of resistance is computed by multi-
plying the one electron value with the total number of electrons. This single-particle
picture works very well in conductivity of bulk systems and Landauer’s theory since
interactions between electrons and nucleus and electrons and electrons are negligible
in high dimension by efficient screening. However, interactions affects electron trans-
port rather significantly in lower dimensions partly because of low electron density and
partly because of insufficient screening among particles. Therefore, single-particle pic-
ture breaks down in lower dimensional conductors and it should take into account of
interactions. It is not a simple task to handle various forms of interactions with many
electrons, especially Coulomb interactions between electrons are notoriously difficult
to be solved in an analytical manner. Such conductors where particle-interactions
cannot be ignored are particularly called ‘strongly correlated systems’.
Besides conductance, noise has also been actively studied in mesoscopic conduc-
tors. The knowledge of noise properties of mesoscopic conductors is invaluable as
quantum limited measurements are performed. Since these empirical results become
the lower bound of performance of quantum information process and quantum compu-
tations [52,53]. There exist different origins to generate fluctuations. Internal micro-
scopic random processes caused by thermal fluctuations, scattering and tunnelling,
quantum effects on noise have been pronounced in mesoscopic conductors [54, 55].
Theories to estimate and experiments to measure such noise in systems have been re-
cently more appreciated since noise is very sensitive to correlations of charge carriers
and scattering mechanisms. In addition, theorists have considered electron trans-
port of mesoscopic systems within quantum coherence transport theory as quantum
1.3. SCOPE OF THE WORK 17
scattering problems. Recent endeavors in this theory are put into gaining a com-
plete statistical analysis of charge transport under the name of “full counting statis-
tics” [56–59]. Conductance and noise from discrete charge carriers correspond to
the first and second moment of characteristic functions in the full counting statistics
respectively. What stimulates the advent of full counting statistics is the fact that
higher moments reveal additional information of systems beyond the first moment.
1.3 Scope of the Work
This dissertation is devoted to investigate correlated electron transport at low tem-
peratures in two one-dimensional ballistic conductors, carbon nanotubes and quan-
tum point contacts. The transport properties exhibit salient features of many-body
electron effects. Chapter 2 first examines two theoretical frameworks in condensed
matter physics to describe how to treat many-body interactions: Fermi liquid theory
and Tomonaga-Luttinger liquid theory. In Chapter 3, in-depth quantum transport to
probe electrical properties of mesoscopic conductors is presented including Landauer-
Buttiker formalism which connects the concepts of conductance and transmission.
Due to increasing attention to intrinsic noise in mesoscopic conductor charge trans-
fer, it discusses the basic knowledge about noise types regarding its origin and roles as
well. Practical strategies and technical description to implement electrical transport
measurements are given in the methodology section, Chapter 4.
Chapter 5 and Chapter 6 are the main context to apply previously described the-
ory and experimental techniques into one-dimensional mesoscopic conductors: single-
walled carbon nanotubes and quantum point contact in two-dimensional electron
gas systems. Two chapters are organized in parallel structures, introducing systems
followed by experiment results and physics interpretations. Interesting many-body
effects disclosed in measurement data are analyzed by various attempts, and the inter-
pretation of results are shown. Summary and outlook of this work concludes Chapter
7.
Chapter 2
Many-Body Physics
The fool collects facts; the wise man selects them.
− John Wesley Powell
The ultimate goal of theoretical physics is to seek a simple way to understand
observations in nature. Beauty of it has lied in the success to model complicated
phenomena elegantly by means of mathematics, a language of physics. The efforts
are driven by a search for microscopic level understandings on natural phenomena in
all physics fields. In particular, condensed matter systems are composed of immense
numbers of particles. To track down all degrees of freedom of all constituents in sys-
tems for microscopic understandings is not only daunting but also impossible. Not to
mention including all sorts of interactions amongst particles. At first, theorists have
attempted to explain, with a few simple and phenomenological parameters, common
classical macroscopic phenomena including transport properties such as phase tran-
sition and resistivity of metals. Statistical mechanics is a consequence of such efforts.
In addition to classical phenomena, quantum phenomena have been included later as
a subject of quantum statistical mechanics. Unlike the classical case, quantum statis-
tical mechanics handles indistinguishable particles. The strategy to treat them is to
use appropriate statistical distribution functions: Bose-Einstein statistics for bosons
and Fermi-Dirac statistics for fermions. In the case of fermions including electrons
18
2.1. CHARGE SCREENING 19
or holes, Pauli exclusion principle regulates their flow. Moreover, it turns out that in
order to understand real electron dynamics, the effect of particle interactions particles
should be taken into account as well together with Fermi-Dirac distribution and Pauli
exclusion principle. Diverse interactions among particles exists including electrostatic
energies between electron and ionized atoms, between electron and lattice vibrations
(phonon), and between electron and electron. How to treat these interactions become
a subtle issue. These interactions come to play crucial roles to exhibit interesting fea-
tures in some systems which are categorized as “strongly correlated systems”. The
examples of such systems are high temperature superconductors and one-dimensional
quantum wires.
This chapter is devoted to introduce two theories which treat interactions in con-
ductors: first, Fermi-liquid (FL) theory which have been successful to exploit bulk
systems for weakly interacting particles; second, Tomonaga-Luttinger liquid (TLL)
which describes interacting one-dimensional conductors whose unique features FL
fails to explain. Before two theories, an important concept is described, “screening”
for electron-electron interactions to validate theories which are applied to specific
systems.
2.1 Charge Screening
Coulomb interactions between charged particles are long-ranged, decaying as a power
law of 1/r, where r is the distance between them. Noticing that people have success-
fully explained physical phenomena in bulk systems without considering such interac-
tions, people have realized that charge screening among many particles attributes to
mitigate the long-range nature of the interactions. In other words, screening causes a
significant reduction of electric field in space. Of many ways to estimate such effect,
an approach introduced below is pedagogical with a link to the elementary electro-
magnetism theory. A basic idea is that the dielectric function is modified due to an
induced electrostatic potential by the induced charge density. Here the argument of
screening is restricted for the static response to long wavelengths in the Thomas-Fermi
approximation [60,61].
20 CHAPTER 2. MANY-BODY PHYSICS
Screening in three dimension
Suppose an induced charge density ρ(~R) of one electron at the origin in three dimen-
sion (3D), ρ(~R) = −eδ(0). The unscreened Coulomb potential Vuns(r) between two
electros far apart by a distance r is
Vuns(r) =e2
4πǫ0ǫbr, (2.1)
in a MKS unit for dielectric constants of vacuum and the media, ǫ0 and ǫb, respectively.
The induced charge polarizes and induces an electrostatic potential, φ(r) satisfying
the Poisson equation
∇2φ(r) =ρ(~R)
ǫ0ǫb= − e
ǫ0ǫb. (2.2)
The Fourier transform of Eq.(2.2) in the reciprocal space ~Q yields
Q2φ = − e
ǫ0ǫb. (2.3)
Therefore, the induced electrostatic potential in the Fourier space is φind = ρind/ǫ0ǫbQ2.
Note that the induced charge ρind is caused by the change in number density of elec-
trons dn such that ρind = −edn and that the total electrostatic potential φtot is due
to the total charge densities including the induced charge as well as external charge
densities. The total electrostatic potential relates to the energy change in the poten-
tial energy dE = −e ˜φtot or the negative change in the Fermi level dE = −dµ for
ρind. These modify the dielectric function in the Fourier space,
ǫ( ~Q) = 1 −˜φind( ~Q)
˜φtot( ~Q)= 1 +
e2
ǫ0ǫbQ2
dn
dµ≡ 1 +
Q2TF
Q2. (2.4)
The Thomas-Fermi screening wavenumberQTF is defined byQTF =√
(e2/ǫ0ǫb)(dn/dµ).
Plugging a new dielectric function into the unscreened potential in the Fourier space
makes the formula of the screened potential Vscr(Q),
˜Vscr(Q) =e2
ǫ(Q)ǫ0ǫbQ2=
e2
ǫ0ǫb
1
Q2 +Q2TF
, (2.5)
2.1. CHARGE SCREENING 21
which is the Lorentzian shape. Converting Eq. (2.5) back into the real space by the
inverse Fourier transform, it gives
Vscr(r) =e2
4πǫ0ǫb
e−QTFr
r, (2.6)
explicitly telling that the long-range Coulomb interaction becomes negligible with an
exponential decay along distance. Therefore, in bulk systems, macroscopic phenom-
ena can be understood without considering Coulomb interactions. Furthermore, at
low temperatures the ratio of dn and dµ is given by the density of states n3D at the
Fermi level, thus a large density of states help the rapid decay of the Coulomb inter-
action in space. As an example, metal has QTF ∼ 1 - 2 A due to a large density of
states at the Fermi level. It explains well why the Coulomb interactions in metal are
insignificant. This result can be understood as follows: repulsive interactions between
electrons leave an electron away from other electron clouds. The emptiness nearby
the electron can be relatively considered as holes, which screen the negative electrons.
Therefore, electrons far away do not feel the potential of these screened electrons: the
more electrons nearby, the more effective screening happens.
Screening in two dimension
Similar to 3D case is the basic idea to compute the screened potential in two-dimension
(2D) (a x − y-plane); however, the difficulty lies in the fact that the induced elec-
trostatic field is in 3D although the charge density is fixed in a 2D plane. Suppose
an induced charge density in 3D , ρ(~R) which is written as ρ(~R) = σ(~r)δ(z) with a
charge density in 2D, σ(~r), at the z = 0 plane. ~R and ~r denote the spatial vector in
3D and 2D. The discontinuity in the electric field at the plane (z = 0) requires the
charge neutrality. In the Fourier space with Gaussian theorem, the induced potential
φind(~q) in 2D is easily computed φind(~q) = ˜σind/2ǫ0ǫbq. The dielectric function in 2D
is written by
ǫ = 1 +e2
2ǫ0ǫbq
dn
dµ≡ 1 +
qTF
q. (2.7)
22 CHAPTER 2. MANY-BODY PHYSICS
The density of states at the Fermi level in 2D at low temperatures is constant at
m/π~2 with a mass m. It means that the value of 2D Thomas-Fermi screening qTF
is independent of the electron density. Once again that the above description is valid
for the static response to the long wavelength.
The unscreened potential in 2D real and momentum spaces with the distance r
are
Vuns(r) =e2
4πǫ0ǫbr
˜Vuns =
∫ ∞
0
Vuns(r)eiqr cos θrdrdθ
=e2
2ǫ0ǫbq.
(2.8)
The screened potential with the dielectric function becomes
˜Vscr =e2
2ǫ0ǫb
1
q + qTF
. (2.9)
Note that the analytical result of the Fourier transform does not exist, but the trend
at large r would be found as [61]
Vscr ∼ e2
4πǫ0ǫb
1
q2TFr
3. (2.10)
The asymptotic behavior in Eq. (2.10) describes that the 2D screened potential
follows the power-law in space. The power-law decay with a exponent of 3 is much
better than the inverse r decay, but it is less sufficient to ignore charge interactions
in 2D compared to the bulk case.
Screening in one dimension
The screening in 1D has even more subtlety since the analytical form of Fourier
transform of V (r) = e2/r does not exist
V (q) =
∫ ∞
−∞dre2
reiqr, (2.11)
2.2. THE FERMI LIQUID THEORY 23
where r is 1D distance between electrons. As a simple attempt, suppose that the 1D
conductor with a uniform line charge density σ is screened by a metal conducting
plane at a distance d from the center. The potential energy between two conductors
is V (r) = (σ/2πǫ0ǫb) ln(Rs/r). In this case Rs is the effective length for screening in
1D. The logarithmic nature of the potential remains in much longer scales, meaning
the screening in 1D is not effective at all in comparison of that in higher counterparts.
Therefore, the understanding of properties in 1D should consider carefully the effect
of interactions among electrons.
In summary, the effectiveness of Coulomb interaction screening according to di-
mensionality can be intuitively understood by geometry. With a negative charge at
the origin, the charge repels electrons lying all possible directions in 3D, thus the
charge is completely screened by relatively positive charges and more electrons are
available at a larger distance. In 1D, on the other hand, only left and right electrons
can be pushed away and this push continuously kicks neighbors, yielding collective
behavior.
2.2 The Fermi Liquid Theory
The Fermi liquid (FL) theory is one of the successful theoretical framework to de-
scribe physical properties of weakly interacting many-body condensed matter systems
such as the liquid state of 3He and conductivity in metals and semiconductors. Before
delving into the details of the FL theory, to remind statistical terminologies among
N-particle systems is of help. There are three classical phases in macroscopic world:
gas, liquid and solid. Of many properties to identify phases, the strength of interac-
tions among particles becomes a good measure. An ideal gas phase appears as mutual
interactions between particles are negligibly smaller than kinetic energy, whereas the
solid state is stable as the interactions between particles are strong. The interme-
diate phase, liquid is between gas and solid. Similarly, in the case of electrons in
transport processes, Fermi gas systems refer to ones in which electrons can be treated
independently because electrons are not interacting each other. Thus, understand-
ing physical properties of such systems can be obtained sufficiently enough by an
24 CHAPTER 2. MANY-BODY PHYSICS
independent single particle.
The Fermi Gas
The knowledge of any systems is complete when the eigenstates (ground state and
excited states) and the elementary excitations of the system are identified. There is
a N-particle Fermi gas system. It is the goal that finds the spectra of eigenstates
and the excitations. The eigenstates of a single particle within a volume (V) are
simple plane waves in the absence of the potential energy. They are written in the
momentum (~k) space in 3D
|~k〉 =1√Vei~k·~r (2.12)
with the quadratic energy dispersion
E~k =(~~k)2
2m(2.13)
where m is the electron mass. The ground state of the system is as the energy
states are occupied below the Fermi energy (EF ). The phase volume of the complete
Fermi sphere relates to the total number of electrons N taking into account of spin
degeneracy,
4πk3
F
3(2π)3
V
=N
2
N = Vk3
F
3π2,
(2.14)
where kF is the radius of the sphere shown in Fig. 2.1(a), and the Fermi energy
EF is defined as EF = (~ ~kF)2/2m. The momentum distribution of the system is the
sharp step function at kF (Fig. 2.1(b)). The Fermi gas system can be excited by
three ways: first, add a particle into an energy state above EF since all states below
EF are fully occupied; second, remove a particle from a state below EF, leaving a
hole inside the Fermi sphere; third, bring a particle below EF into a state above EF.
2.2. THE FERMI LIQUID THEORY 25
kF
ky
kxkFkF
kFkF
kyky
kxkx
(a)
(c)
2kF
E
q
(c)
2kF2kF
E
q
(b)
n~kn~k
kFkF
1
Figure 2.1: (a) The Fermi sphere of non-interacting electron systems. (b) The momen-tum distribution function of the Fermi gas systems. (c) The spectrum of particle-holeexcitations in the Fermi gas systems.
26 CHAPTER 2. MANY-BODY PHYSICS
There are respectively named as particle excitations, hole excitations and particle-
hole excitations. The particle-hole excitations are illustrated in Fig. 2.1(c), noting
that the momentum of q due to scattering between a particle and a hole is continuous.
The Fermi Liquid
In the FL system, interactions between electrons need to be considered unlike in the
Fermi gas. Landau approached the FL system from the Fermi gas with a hypothesis,
an adiabatic switch-on of interactions. Further he considered the long wavelength
limit, i.e. low energy excitations near the Fermi energy. Therefore, the eigenstates
of the Fermi gas and the FL are in one-to-one correspondence [?, 62]. He captured
the idea that the interactions would modify the energy dispersion relation of the
Fermi gas, consequently changing the mass of electrons in the system. Introducing
the effective mass m∗ which reflects the strength of mutual interactions, Landau
established the FL theory within the single particle picture. The ground state of the
FL system is still the Fermi surface. The FL theory again studies the elementary
excitations of the system.
The essence of the FL theory is the existence of quasi-particles. They are low-lying
elementary excitations and they consist of electrons with the density fluctuations aris-
ing from the particle interactions. Due to the fact that quasi-particles are formed from
electrons, they also obey fermionic commutation relations. The formal description of
quasi-particles is the Green’s function of electrons:
G(~k, ω) =1
E0(~k) − ω − Σ(~k, ω), (2.15)
where E0 is the energy dispersion of the Fermi gas and Σ is the self-energy due
to many-body interactions. The pole of the above Green’s function provides the
excitation energy, representing quasi-particles.
With the self-energy term, many quantities are derived including the effective mass
m∗ of quasi-particles, the quasi-particle renormalization factor Z and the spectral
density A which describes the probability of finding a certain state [63]. m∗ and Z
2.2. THE FERMI LIQUID THEORY 27
are expressed in terms of the self energy as follows:
m∗ = m
(
1 − ∂Σ
∂ω
)(
1 +m
kF
∂Σ
∂k
)−1
(2.16)
Z =
(
1 − ∂Σ
∂ω
)−1
. (2.17)
n~k
kF
1
Z
n~kn~k
kFkF
1
Z
Figure 2.2: The momentum distribution of the Fermi liquid system.
Physically, Z indicates the amplitude that electrons remain as quasi-particles. In
the momentum distribution, the perfect step function is modified by Z in Fig. 2.2.
As Z closes to 1, the system is less interacting. The stronger interactions, the degree
of smearing is larger. The analytical computation of the self-energy regarding the
particle interactions is based on an assumption that the interactions are short-range.
Charge screening discussed in the previous section provides a key to validate the
FL theory in higher dimensions since the effective screening reduces the long-range
Coulomb interactions among electrons. Therefore, the FL theory works well to de-
scribe transport processes in systems whose interactions are short range and isotropic
such as metals, semiconductors and liquid 3He.
28 CHAPTER 2. MANY-BODY PHYSICS
2.3 The Tomonaga-Luttinger Liquid Theory
The successful FL theory fails in 1D. The FL theory breakdown in 1D conductors
can be understood roughly in terms of inefficient charge screening in 1D mentioned
earlier. The previous section describes that the long-range Coulomb interactions
survive, correlating electrons in 1D since any excitation at a particular site spreads
over the whole lattice like the domino effect. The collectiveness is the unique feature of
1D excitations, meaning that Landau’s quasi-particles do not exist [64]. The absence
of the quasi-particles in 1D is from the multiple poles of the Green’s function [64]. A
rigorous attempt to describe 1D electron gas systems is formulated in the Tomonaga-
Luttinger liquid (TLL) theory. Tomonaga and Luttinger came up with an exactly
solvable model in 1D with insights that collective modes are bosonic nature and the
linearization of the dispersion near the Fermi level gives low energy properties [62,64].
Bosonization
The Fermi surface of 1D is two points at ±kF. Particle-hole excitations in 1D are only
possible near q = 0 or q = 2kF nearby the Fermi points, whereas any q values below
2kF are allowed for the particle-hole excitations in higher dimensions by conserving
the energy and momentum. In the limit of q → 0 and ω → 0, the excitation spectrum
is linear, resembling a phonon mode. This resemblance hints that the Hamiltonian
of 1D electron gas system can be derived by phonon displacements as a rather in-
tuitive approach. In the extreme limit where the interactions are stronger to form
Wigner crystal, the ground state of such 1D system can be modelled as an equally
spaced particle chain. The lattice constant between particles are a and it is inversely
proportional to the 1D electron density n0, i.e. a = 1/n0. The phonon-like lowest
excitations are written by the displacement ri
ri = r0i +
a
πθi, (2.18)
r0i is the initial displacement and θi is the displacement variable. As one electron
propagates to a nearest neighbor site, the variable advances by π. By definition, the
2.3. THE TOMONAGA-LUTTINGER LIQUID THEORY 29
kinetic energy (K) is derived from the displacement definition:
K =∑
i
1
2mr2
i =
∫
dxma
2π2θ(x)2. (2.19)
If we assume the 1D Coulomb interaction is screened by a ground plane at Rs where
Rs ≫ a, the effective potential V0 is V0 =∫
dx e2/x = 2 e2 log(Rs/a). With this
effective potential, the short-range interaction becomes feasible. If one additional
electron sits inside the chain, the overall change in θ is −π. Thus, the electron
density change δn(x) relates to the change of θ in x such that δn(x) = - ∂xθ(x)/π.
The potential energy (U) is for a short-range interaction
U =
∫
dxdx′1
2V (x− x′)δn(x)δn(x′)
=
∫
dxV0(δn(x))2
=
∫
dxV0
2π2(∂xθ(x))
2.
(2.20)
Therefore, the Hamiltonian density H is given by
H =ma
2π2(∂tθ(x))
2 +V0
2π2(∂xθ(x))
2. (2.21)
Substituting g =√
π~vF/V0 and vρ =√
V0/ma = vF/g, Eq.(2.20) becomes
H =~
2π
[
1
vF
(∂tθ)2 + vρ(∂xθ)
2
]
. (2.22)
Consider a spinless 1D system. The low energy properties can be studied by
linearized dispersion relation near the Fermi level. The positive (negative) slope at
kF(-kF) corresponds to right (left) moving channels. The Hamiltonian H is computed
by integrating the previous Hamiltonian density H =∫
dxH. It consists of the kinetic
energy part and a repulsive Coulomb potential energy with a strength constant λ for
30 CHAPTER 2. MANY-BODY PHYSICS
low energy excitations
H = −ivF
∫
dx[
ψ†R∂xψR − ψ†
L∂xψL
]
+ λ
∫
dx(ψ†RψR + ψ†
LψL)2. (2.23)
ψi are field operators for left(L) and right(R) moving electrons. These fields are
expressed by the bosonic fields φ and θ with the cut-off constant Λ such as
ψR(L) =1√2πΛ
ei(φ±θ). (2.24)
Thus, Eq. (2.22) is rewritten by the bosonic fields [65,66]
H =~vF
2π
∫
dx
[
(∂xφ)2 + (∂xθ)2) +
λ
π~vF
(∂xθ)2
]
≡ ~vF
2π
∫ [
(∂xφ)2 +1
g2(∂xθ)
2
]
,
(2.25)
with the renormalization factor g = (1 + λ/π~vF )−1
2 .
The Tomonaga-Luttinger Liquid Parameter g
The TLL parameter, g is a measure of the interaction strength, defined as a dimen-
sionless quantity,
g =
(
1 +V0
π~vF
)− 1
2
, (2.26)
for a interaction potential V0. The second term in Eq. (2.26) indicates the competition
of the potential and kinetic energy in a system. In the absence of V0, g becomes 1,
recovering the non-interacting Fermi gas system, whereas V0 > ~vF > 0 for a repulsive
Coulomb interaction leads to g < 1. The stronger interactions V0, the smaller g. Note
that g can be also greater than 1 if attractive Coulomb interactions are dominant
among particles. The TLL parameter g emerges in various 1D properties such as
the fractional charge ge, the charge mode velocity vF/g, and the power-exponents of
correlation functions.
2.3. THE TOMONAGA-LUTTINGER LIQUID THEORY 31
Single-Walled Carbon Nanotubes
Single-walled carbon nanotubes (SWNTs) are one specific example of one-dimensional
conductors. Metallic SWNTs have been predicted as the TLL system [65, 66]. The
transport properties in the tunnelling regime, where tubes are isolated from metal
reservoirs, exhibited the TLL features as the power-scaling conductance by means of
the bias voltage and the temperatures [23]. Recently, angle-integrated photoemission
measurements obtained the spectral function from SWNT mats, claiming the direct
observation of the TLL features in SWNTs [67]. The search of the TLL behavior in
SWNTs is active since the strongly correlated SWNTs serve as a basic ingredient of
quantum electron entanglers [68–71].
The lowest bands of metallic SWNTs in the Brillouin zone (BZ) are linear. Since
the hexagonal BZ of SWNTs contains two inequivalent K and K ′, four bands are
at the same energy reflecting the orbital and spin degeneracies. The Hamiltonian
HSWNT,0 of an infinite SWNT without interactions can be derived in a similar way
with bosonic fields with a band index i= 1,2 and a spin index σ =↑, ↓
HSWNT,0 =∑
i,σ
∫
dx[
i~vF (ψ†Riσ∂xψRiσ) − ψ†
Liσ∂xψLiσ
]
. (2.27)
Similar to the above the electron fields are expressed by bosonic fields, φRiσ(Liσ) =
(1/√
2πΛ)ei(φiσ+θiσ). The bosonic fields obey the commutation relation [φiσ(x), θjσ′(x′)] =
−iπδijδσσ′Θ(x − x′). Known the fact that the interaction term relates to the total
charge density, new bosonic fields are defined for convenience such that the total
charge and spin fields are denoted as θic(s) = (θi↑±θi↓)/√
2 and θc(s)± = (θ1µ±θ2µ)/√
2.
φ fields are defined similarly. The new fields denoting as a = (c(s),±) obey the same
commutation relations [φa(x), θb(x′)] = −iπδabΘ(x − x′). These fields convert the
free Hamiltonian into the bosonized form of H with one charged excitation and three
neutral excitations [65, 66]. In an infinite SWNT, the possible scattering process
is repulsive forward scattering which requires a small momentum transfer q ∼ 0,
whereas backscattering processes are allowed between two branches with a big mo-
mentum transfer q ∼ 2kF. The forward scattering arises from a long-ranged repulsive
32 CHAPTER 2. MANY-BODY PHYSICS
Coulomb interaction. The Hamiltonian of this part is written as Hint =∫
dxV0ρ2tot.
Assume that the Coulomb interaction is screened by the ground plane at Rs, the
effective screened potential is V0 = e2 ln(Rs/R) for a SWNT radius R. Note that the
interactions only change the total charge part, making the total Hamiltonian HSWNT
to be
HSWNT =vc
2π
∫
dx
[
g(∂xφc+)2 +1
g(∂xθc+)2
]
+∑
a=(c−,s+,s−)
vF
2π
∫
dx[(∂xφa)2 +(∂xθa)
2].
(2.28)
The propagating velocity of the total charge mode vc is faster than the Fermi velocity
by 1/g.
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Rs/R
g
Figure 2.3: The TLL parameter g as a function of Rs/R. Two red lines are markedat g = 0.2 and g = 0.3.
The TLL parameter g for the SWNT is written in the CGS unit as
g =
[
1 +8e2
π~vF
ln
(
Rs
R
)]− 1
2
. (2.29)
2.4. SUMMARY 33
Figure 2.3 plots g vs Rs/R with vF = 8 × 107 cm/s. It shows that the g value falls
quickly between 0.2 and 0.3 for Rs/R > 5. The logarithmic dependence on Rs/R
explains that the value of g is rather insensitive to the actual value of Rs [65]. The
small value of g indicates that strong Coulomb interaction is present in SWNTs.
2.4 Summary
Chapter two has presented the background level of theoretical aspects regarding elec-
tron transport. The important concept “screening” has been examined in terms of di-
mensions. In higher dimensional systems, weakly interacting (or non-interacting) FL
theory has been a successful description for electrical properties, whereas in the lower
dimensional conductors, especially one-dimensional cases, there has been observed
that many quantities and phenomena are beyond the FL theory, where interactions
among charged particles are not negligible due to insufficient screening. The TLL
theory particularly focuses on the 1D system and as an example, the application of
the TLL on single-walled carbon nanotubes has been introduced. Such systems are
in the correlated transport regime, which is the main theme of this thesis. The next
chapter will discuss much narrower topics but more practical observable quantities
:conductance and shot noise.
Chapter 3
Mesoscopic Electron Transport
I know of no other advice than this:
Go within and scale the depths
of your being from which your very life
springs forth.
− Rainer Maria Rilke
The preceding chapter discussed theoretical aspects of many-body physics and in-
troduced two backbone descriptions for weakly and strongly interacting many-electron
condensed matter systems: Fermi liquid theory and Tomonaga-Luttinger liquid the-
ory. Both theories have nicely established a mathematical framework to study the
ground state properties and elementary excitations in terms of second quantization
techniques. Furthermore, they link correlations between system observables in equi-
librium and in non-equilibrium situations to scattering and transport processes.
Transport processes is related to the response of a system to external stimulation.
For example, firing up a part of a system generates a temperature gradient, inducing
a net heat flow across it. Closely connected to the thermal conduction, the movement
of electrons due to a non-zero electrical potential along a system yields electrical con-
ductivity, one of the material characteristics. Conductivity measurements by probing
a current change according to a bias voltage across a system, have provided valuable
34
3.1. LINEAR RESPONSE THEORY 35
information to identify the states of matter, metal or insulator. This chapter pays
a particular attention to the electrical transport properties, discussing fundamentals
and the implications in mesoscopic conductors.
3.1 Linear Response Theory
Linear response theory (LRT) raises a practical question: how a system in equilibrium
responds as its equilibrium state is disturbed. It formulates the response function of
a many-particle system which is stimulated by an external source. LRT assumes that
the external stimulation is weak enough that it can be treated as a perturbation,
justifying the Taylor series expansion. Plus, the perturbation expansion series are
converging rapidly after the first linear term; thus, considering the first non-trivial
linear term would be sufficient to describe the response of systems. This response
function is a measurable quantity, therefore it is real-valued. In transport, the re-
sponse function is a macroscopic transport coefficient. Since it is shown that the
response function relates to the correlation functions in the system, LRT describes a
nonequilibrium system in terms of fluctuations about its equilibrium state. There-
fore, understanding the dynamics of a system in equilibrium is essential to predict
nonequilibrium situations.
Suppose a system whose isolated Hamiltonian is denoted as H0. If a weak time-
dependent disturbing field A ·F (t) is applied to the system at time t0, the perturbed
Hamiltonian H at later time t becomes H = H0 − A · F (t) where A is the inter-
nal quantity conjugate to the field F (t). LRT says that the average of A in the
nonequilibrium 〈A(t)〉 can be written as
〈A(t)〉 = 〈A(t)〉0 +
∫ t
−∞dt′R(t, t′)F (t′) +O(F (t)2), (3.1)
where 〈...〉0 is the average over equilibrium ensembles. R(t, t′) is the linear response
function, which relates two times t′ and t. t′ is the time at which the external
field acts on the system and t is the time of measurement. Thus t > t0, it is the
causality property. A simple example of the response function is the conductivity in
36 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
equilibrium, which in connection with Eq.(3.1) is known as the Green-Kubo formula.
It states that the equilibrium conductivity σ of a one-dimensional system subjected
to a constant voltage V at time t = 0 is given in terms of the current density jx(t)
along the x-direction,
σ =V
kBT
∫ ∞
0
dt〈jx(0)jx(t)〉0.
Conductance in mesoscopic conductors can be computed as the response function
described above. Note that the mathematical strategies are different depending on
which regime (either ballistic or diffusive) the actual transport occurs.
3.1.1 Ballistic Transport
Ballistic transport refers to the transport of electrons without encountering any types
of scattering sources. In other words, the system size is smaller than the mean free
path and the inelastic scattering length. Based on the point of view that conductance
arises from scattering, conductance is predicted to be infinite in the ballistic regime;
however, finite conductance has been measured in the ballistic conductors. This ob-
servation boosted theoretical interests to understand the origin of finite conductance.
(a) (b)c
E
k
E
k
(a) (b)c
E
k
E
k
(b)cc
E
k
E
k
Figure 3.1: (a) An one-dimensional ballistic conductor in a two-terminal configura-tion. (b) The energy dispersion of free electrons in a reservoir (left) and a conductor(right).
3.1. LINEAR RESPONSE THEORY 37
Laudauer-Buttiker Formalism
Landauer brilliantly captured the wave nature of electrons in mesoscopoic conductors,
and he interpreted the conductance as the transmission probabilities of propagating
modes analogous to electromagnetic fields in optical waveguides.
Suppose a simple one-dimensional (1D) ballistic conductor with two leads coupled
to bulk electron reservoirs, as illustrated in Fig. 3.1(a). Adiabatic transition from bulk
reservoirs to the device and zero temperature are assumed. In case of free electrons,
Fig. 3.1 (b) presents the energy dispersion relations in bulk reservoirs (left) and the
conductor (right). The horizontal axis represents the longitudinal wavenumber k. Due
to the spatial confinement, the allowed modes in the conductor are discrete, while the
modes in the bulk are relatively dense. Therefore, not all modes below the Fermi
energy can propagate into the conductor due to energy and momentum conservation,
yielding that only certain modes can be matched in both regions. Mode reflection
at the interface of two dissimilar materials causes finite conductance even with a
ballistic conductor. Sometimes this finite resistance is called ‘contact resistance’. In
the simplest case, one channel in the conductor exists. The current I across the
conductor with the applied bias voltage V is given as I =∫ EF +eV
EFeρ(E)vg(E) with
energy-dependent density of states and group velocity. The density of states ρ in
1D is given ρ = 1/2π~vg(E). Note that in 1D, there is a magic cancellation of the
velocity component, yielding the product of ρ(E) and vg(E) is constant 1/h. Thus,
the current including spin degeneracy is
I =
∫ EF +eV
EF
e2
hdE =
2e2
hV,
reducing the conductance G to G = I/V = 2e2/h ≡ GQ denoted as the spin-
degenerate quantum unit of conductance. GQ is measured when the mode is com-
pletely transmitting into the opposite reservoir. For a mode which is transmitting
with a probability T , the conductance G is G = GQT . Moreover, if there are more
than one channel involved in the transport process and each mode has an individual
transmission probability Ti, then the conductance G is obtained as a sum over all
38 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
modes
G = GQ
∑
i
Ti,
known as the ‘Landauer formula’ [72, 73]. Then, Buttiker further extended the Lan-
dauer’s formula into multi-lead and multi-mode systems even in presence of a mag-
netic field. He established coherent scattering formalism. Conductance measured in
two leads α and β is ,
Gα→β = GQ
Nα∑
n=1
Nβ∑
m=1
|tβα,mn|2,
with channel modes m,n [74].
Scattering Matrix and Transmission Matrix
a1
b1
(a) (b)
±°
® ¯a2
b2
a1a1
b1b1
(a) (b)
±°
® ¯
(b)
±±°°
®® ¯a2a2
b2b2
Figure 3.2: A two-port system represented by second quantized operators.
Second quantization representation is an elegant way to describe mesoscopic con-
ductors. This is powerful in many aspects: first, it deals straightforwardly with
indistinguishable many particles; second, it automatically satisfies exchange rules of
bosons or fermions. The previous 1D, one-channel conductor is regarded as a two-port
system drawn in Fig. 3.2(a). The operators ai annihilate particles in the incoming
channels into the scattering site, and the operators bi do particles in the outgoing
3.1. LINEAR RESPONSE THEORY 39
channels. The index i is either 1 or 2. How incoming and outgoing operators are
related is written in a compact matrix form. There are two different ways to connect
those operators: (1) scattering matrix S and (2) transfer matrix T . A certain form
is more efficient than the other, appropriate for situations. The components of these
matrices are transmission and reflection coefficients between corresponding modes.
Consider an example to view how to form the matrices with a two-port system
with one channel. First, the S-matrix gives an obvious connection of the incoming
channels versus the outgoing channels such that
(
b1
b2
)
=
(
r11 t12
t21 r22
)(
a1
a2
)
≡ S
(
a1
a2
)
. (3.2)
Since S is unitary, i.e. SS† = S†S = 1, two conditions among components should be
met:
SS† =
(
r11 t12
t21 r22
)(
r∗11 t∗21
t∗12 r∗22
)
=
(
1 0
0 1
)
, (3.3)
reading that
|r11|2 + |t12|2 = |r22|2 + |t21|2 = 1
r11t∗21 + t12r
∗22 = r∗11t21 + t∗12r22 = 0.
(3.4)
Second, a T-matrix describes how the left operators propagate to the right side:
(
b2
a2
)
=
(
T11 T12
T21 T22
)(
a1
b1
)
≡ T
(
a1
b1
)
. (3.5)
The benefit of the T-matrix representation is to readily compute the overall T-matrix
Tall as a particle propagates several T-matrices until it reaches the final location.
Explicitly, it means
(
bN
aN
)
= T (N)T (N−1) · · · T (1)
(
a1
b1
)
≡ Tall
(
a1
b1
)
. (3.6)
40 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
Thus, Tall expresses Tall = T (N)T (N−1) · · ·T (1). The components of T are rewritten in
terms of rij and tij,
T11 = t21 −r11r22t12
,
T12 =r22t12,
T21 = −r11t12,
T22 =1
t12.
(3.7)
3.2 Fundamentals of Noise
The term ‘Noise’ in electronic circuits has dominantly negative meanings since it of-
ten refers to spurious, unwanted uncontrollable contamination to a signal. Therefore,
noise, in other words signal fluctuations, is a limiting factor of the sensitivity of equip-
ments and apparatus. Electronic systems are vulnerable to external noise sources via
magnetic, capacitive, radio frequency couplings [75]. These couplings ruin main sig-
nals unless they are well taken care of in right manners. Grounding and shielding are
primary solutions to fight against external noise sources [76]. In addition to cumber-
some degradations, there exist other types of noise, so called intrinsic noise of systems.
These noise sources cannot be eliminated by grounding and shielding techniques, and
they reflect statistical features arising from enormous numbers of electrons in trans-
port processes. This section is devoted to introduce some background knowledge of
these intrinsic noises.
3.2.1 Fluctuation and Dissipation Theorem
Statistical mechanics is a study of many-body systems, asking the effective way to
treat countless degrees of freedom [77]. It has been successful to describe macro-
scopic thermodynamic phenomena in equilibrium. It has been investigating the non-
equilibrium and irreversible processes as well. Of numerous approaches, the fluctua-
tion and dissipation theorem (FDT) is of importance in that it basically tells that the
3.2. FUNDAMENTALS OF NOISE 41
non-equilibrium properties are closely related to the equilibrium quantities [78]. In
detail, it provides a general relationship between the response of a system disturbed
by an external source and the internal fluctuations of the system without the distur-
bance. The validity of FDT lies in the linear response regime, indicating that the
external disturbance is weak and the dominant term is the linear one. The response
of the system is often characterized by a response function, for example admittance
or impedance in the electronic circuits. On the other hand, the internal fluctuations
reflect correlation functions of physical quantities in thermal equilibrium. Therefore,
the roles of the FDT can be summarized into two: first, the FDT predicts the fluctua-
tion characteristics or intrinsic noise of the system from the known properties; second,
the FDT provides a basic formula to derive the known properties such as resistance
from the analysis of fluctuations in the system.
Brownian motion is a prototypical phenomenon to provide insight into the FDT,
describing random behavior of objects. And it results in statistical fluctuations in
thermal equilibrium systems. For convenience, consider a one-dimensional system
that moves in one direction x with velocity v(t) at time t. The system is not com-
pletely isolated from the external world, but only couples with the outside environ-
ment weakly. The slowly varying external coupling is given as F(t), whereas F (t), the
interaction of the system with other degrees of freedom, is rapidly fluctuating. The
latter sets one time scale, the ‘correlation time’ τ ∗, which measures roughly the mean
time between maxima of F (t). For macroscopic times τ , i.e. τ ≫ τ ∗, the equation of
motion is
mdv
dt= F(t) + F (t). (3.8)
After the integration over τ followed by the ensemble average, Eq.(3.8) is rewritten
as
m〈v(t+ τ) − v(t)〉 = F(t)τ +
∫ t+τ
t
〈F (t′)〉dt′. (3.9)
The integrand 〈F (t)〉 in the second term on the right side is associated with an energy
change ∆E in the external world at temperature T such that
〈F (t)〉 =1
kBT〈F (t)∆E〉0, (3.10)
42 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
with the ensemble average 〈....〉0 in equilibrium state [77]. Note that the energy
change in the external world is equivalent to the negative work done by the force
F (t),
∆E = −∫ t′
t
dt′′v(t′′)F (t′′) ≈ −v(t)∫ t′
t
dt′′F (t′′). (3.11)
The approximation in the last equivalence is valid since the system velocity vary
negligibly over τ . Plugging the last equation into Eq. (3.10), the mean of F (t) is
written with β = (kBT )−1 as
〈F (t′)〉 = −β〈F (t)v(t)
∫ t′
t
dt′′F (t′′)〉0
= −βv(t)∫ t′
t
dt′′〈F (t′)F (t′′)〉0.(3.12)
Since what physically matters is the time change, dummy variables can be changed
accordingly s ≡ t′′ − t′. Equation (3.9) reads
m〈v(t+ τ) − v(t)〉 = F(t)τ − βv(t)
∫ t+τ
t
dt′∫ 0
t−t′ds〈F (t′)F (t′ + s)〉0. (3.13)
In statistical mechanics, 〈F (t′)F (t′ + s)〉0 is referred to as “ correlation function ” of
F (t). Note that the second term on the right side leads to ‘dissipation’ in the system.
3.2.2 General Formulation of Noise
The internal fluctuations in the time domain look random, thus it is quite challenging
to extract quantitative information. Therefore, often the time-information of the
fluctuations is converted into its conjugate parameter, frequency. The frequency
response gives the spectral content of the fluctuations. The quantity in the frequency
domain for correlation functions is called “spectral density”, J(ω). And the relation
between two quantities is via Fourier transformation,
J(ω) =1
2π
∫ ∞
−∞〈F (t′)F (t′ + s)〉0eiωsds,
3.2. FUNDAMENTALS OF NOISE 43
this relation is especially called as “Wiener-Khintchine Theorm”.
Particularly in electronic circuits, the variables which fluctuate according to ex-
ternal stimulations are current or voltage. The Fourier components of fluctuating
currents or voltages are named “power spectral density” because the unit is either
[A2/Hz] or [V2/Hz]. Explicitly, the current noise spectral density SI and the voltage
noise spectral density are expressed in terms of current and voltage operators, ˆI(t)
and ˆV (t),
SI(ω) =
∫ ∞
−∞〈 ˆI(t), ˆI(0)〉eiωtdt
SV (ω) =
∫ ∞
−∞〈 ˆV (t), ˆV (0)〉eiωtdt.
(3.14)
3.2.3 Classification of Intrinsic Noise
10-25
2
4
6
10-24
2
4
6
10-23
2
4
6
10-22
SI
(A2/
Hz
)
100 10
3 106 10
9 1012 10
15
Frequency (Hz)
1/f noise
Shot noise
Thermal noise
Quantum noise
10-25
2
4
6
10-24
2
4
6
10-23
2
4
6
10-22
SI
(A2/
Hz
)
100 10
3 106 10
9 1012 10
15
Frequency (Hz)
1/f noise
Shot noise
Thermal noise
Quantum noise
Figure 3.3: Spectrum of dominant noise sources in frequency domain.
44 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
There are several sources causing fluctuations in time-varying currents: thermal
noise, shot noise, 1/f noise, and quantum noise. In terms of frequency behavior,
two distinct trends are identified. Some noise is constant over some frequency range,
whereas other noise diverges as frequency is decreased. These are often named by
colors such that the former case is white nose and the latter one is pink noise. The
examples of white noise are thermal and shot noise, and the 1/f noise belongs to
the pink noise category. In addition, the noise grows as frequency increases, and
quantum noise falls into this type. Figure 3.3 exhibits the spectra of intrinsic noise
sources. This subsection describes the origin of the intrinsic noise sources and their
mathematical formulations.
Johnson-Nyquist Noise
At finite temperature, the electron collisions with the lattice vibrations and impuri-
ties induce fluctuations in the electrons’ velocity and position. The electronic motion
is well explained by a Brownian particle model. The net microscopic electronic ther-
mal agitation drives a fluctuating voltage across the resistor. Since it exists without
voltage applied to system, it is equilibrium quantity. The microscopic thermal fluc-
tuations provide correlations physical quantities either current or voltage, and they
are associated with dissipative term in the circuit, conductance or resistance. Such
outcome is called the Johnson-Nyquist noise at non-zero temperature of statistical
objects. It is one of prototypical example to which FDT applies explicitly.
R L
I (t)
V0 (t)+
-R L
I (t)
V0 (t)+
-
Figure 3.4: A parallel resistor-inductor circuit.
3.2. FUNDAMENTALS OF NOISE 45
Consider a parallel resistor-inductor circuit drawn Fig 3.4. R is the resistance and
L is the inductance. Given the voltage across the resistor V0(t), the Kirchhoff voltage
law tells that LdI(t)/dt = V0(t), where I(t) is the current of the circuit. Suppose
V0(t) decomposes into a slowly-varying part V (t) and a rapidly-varying part v(t).
Note that in this circuit,there are three relevant time constants: τ ,τm,τ ∗. τ is the
time constant for the macroscopic quantities (e.g.I(t)) of the circuit to change, τm is
the time constant for a change in the electron’s velocity (momentum), and τ ∗ is the
mean-free time between successive collisions of an electron with the lattice. These
three are assumed to obey the following inequality: τ ≫ τm ≫ τ ∗. The slowly-varying
part V (t) changes on the time scale τ and the rapidly-varying part v(t) changes on the
time scale τ ∗. Here, V (t) acts to keep I(t) around I(t) = 0,in other words, V (t) is a
relaxation term for the departure of I(t) from its steady-state value. Upon the above
time scales, the equation of motion, the ‘Langevin equation’ can be derived from the
Kirchhoff voltage law. Now both I(t) and V0(t) can be decomposed into the slowly-
varying part and the rapidly-fluctuating one: I(t) = I(t) + i(t); V0(t) = V (t) + v(t).
Over τm, rapidly-fluctuating components can be averaged out, you can write the
Kirchhoff voltage law as
LdI(t)
dt= V (t).
Considering that V (t) is a restoring force for the steady-state value of I(t) with a
coefficient, R, then
V (t) = −RI(t).
If we approximate I(t) = I(t)+ i(t) ≈ I(t), neglecting the small modulating signal in
current, then the equation becomes
LdI(t)
dt= V (t) + v(t)
≈ −RI(t) + v(t).
(3.15)
Putting γ = R/L, the Langevin equation is re-written as
dI(t)
dt+ γI(t) =
1
Lv(t).
46 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
As a mathematical trick, multiplying eγt and rearranging the terms yield
d
dt[eγtI(t)] =
1
Leγtv(t). (3.16)
Integrating Eq.(3.16) from −∞ to 0,
I(0) =1
L
∫ 0
−∞eγtv(t)dt. (3.17)
Suppose the voltage fluctuation v(t) is stationary. Its covariance function depends on
the time difference so that v(t′)v(t′′) = v(t′ − t′′)v(0) for a stationary process.
I(0)2 =1
L2
∫ 0
−∞
∫ 0
−∞eγ(t+t′)v(t)v(t′)dtdt′,
=1
L2
∫ 0
−∞
∫ 0
−∞eγ(t+t′)v(0)v(t− t′)dtdt′.
(3.18)
Then, changing the integration variables to s = t′ + t′′ and s′ = t′ − t′′, the integral
in Eq. (3.18) becomes,
I(0)2 =1
L2
∫ 0
−∞eγsds
∫ 0
−∞v(0)v(s′)ds′
=1
L2
1
γ
1
2
∫ ∞
−∞v(0)v(s′)ds′
(3.19)
Inserting I(0)2 = kBT/L from the equipartition theorem at temperature T , Eq. (3.19)
iskBT
L=
1
L2
L
R
1
2
∫ ∞
−∞v(0)v(s′)ds′. (3.20)
After replacing a dummy variable s′ by t,the FDT for thermal noise can be written
as
R =1
2kBT
∫ ∞
−∞v(0)v(t)dt. (3.21)
Basically, Eq. (3.20) indicates that the microscopic voltage correlations due to ther-
mal random motions of electrons are related to equilibrium resistance, dissipative
3.2. FUNDAMENTALS OF NOISE 47
term in the circuit. The Wiener-Khintchine theorem is expressed such that
Sv(ω) = 4
∫ ∞
0
v(0)v(t) cos(ωt)dt. (3.22)
Since v(0) and v(t) are correlated only over τ ∗, it would be fair to approximate
ωτ ∗ ≪ 1, leading cos(ωt) ≈ 1. Combining Eq.(3.21) and Eq.(3.22), the Johnson-
Nyquist thermal noise is expressed as
Sv(ω) = 4
∫ ∞
0
v(0)v(t) cos(ωt)dt
≈ 2
∫ ∞
−∞v(0)v(t)dt
= 4kBTR.
(3.23)
Note that the expression does not have frequency dependence, indicating it is white
up to a cut-off frequency. Similarly, the Johnson-Nyquist thermal noise in currents
can be derived such that SI(ω) = 4kBTG, with conductance G. The way to remove
thermal noise is to reach the absolute zero temperature, thus cooling the system would
reduce the amount of thermal noise in systems in a linear manner.
1/f Noise
The 1/f noise has several alternative name, for example, flicker noise, pink noise
or telegraph noise. However, often the 1/f noise or low-frequency noise are rather
conventionally used due to the frequency trend. As name indicates, it is the dominant
noise sources in low frequency. In the circuit, the following equality holds valid with
C1/f , a measure of the relative noise of the sample,
SI(f)
I2=SV (f)
V 2=SR(f)
R2=SG(f)
G2=C1/f
f. (3.24)
Experimentally the above quantities Si(f) are proportional to fα where the power
exponent is roughly α = −1.0±0.1 [79]. The microscopic origin of this noise is unclear
and under investigation [80]. Of many theoretical speculations, a random telegraph
48 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
signal would lead to 1/f noise in that trapping centers in system capture and release
electrons or holes in a random fashion. The question remains what are candidates
of trapping centers. Defects on surface or interfaces act as such centers. Consider a
generation-recombination noise whose physical quantity is X its fluctuation is ∆X.
Suppose ∆X decays with time scale τ [79]. The decay differential equation is
−d∆X(t)
dt=
∆X(t)
τ. (3.25)
The integration of the above differential equation gives solution that
∆X(t) = ∆X(t0)e− 1
τ(t−t0).
The correlation function of X is
ϕX(t) = 〈∆X(t0)∆X(t0 + t)〉 = 〈∆X(t0)∆X(t0)e− t
τ 〉= 〈(∆X)2〉e− t
τ .(3.26)
Plugging the expression of the correlation function into the Wiener-Khintchine theo-
rem, the power spectral density SX(f) is given
Sx(f) = 4
∫ ∞
0
ϕX(t) cos(2πft)dt
= 4
∫ ∞
0
〈(∆X)2〉e− tτ cos(2πft)dt
= 4〈(∆X)2〉[
τ
1 + (2πft)2
]
.
(3.27)
This independent-electron process yields Lorentzian spectrum, exhibiting that it is
white for fτ ≪ 1 while it shows 1f2 for fτ ≫ 1. If there are a large number of
Lorentzian spectra, the overall noise is from a summation of independent process.
Mathematically, it can be computed with a weighting factor g(τ) , which is inversely
proportional to a time τ . Meaning that the Lorentzian spectra have relaxation time
τ satisfying the inequality τ1 < τ < τ2, g(τ) can be written as g(τ)dτ = 1ln(
τ2τ1
)1τdτ .
3.2. FUNDAMENTALS OF NOISE 49
With this g(τ), the spectral density SX(f) becomes
SX(f) =
∫ τ2
τ1
g(τ)〈(∆X)2〉 4τ
1 + (2πfτ)2dτ
=4
ln( τ2τ1
)〈(∆X)2〉
∫ τ2
τ1
1
1 + (2πfτ)2dτ
=2
π ln( τ2τ1
)
(
1
f
)
〈(∆X)2〉[
tan−1(2πfτ2) − tan−1(2πfτ1)]
.
(3.28)
There are three regimes: (1) For f < 1/2πτ2, Eq.(3.28) becomes SX(f) = 4τ2〈(∆X)2〉/ ln(τ2/τ1), which is constant in frequency; (2) For 1/2πτ2 < f < 1/2πτ1, Eq. (3.28)
becomes SX(f) = 〈(∆X)2〉/ ln(τ2/τ1)f , which is 1/f spectrum; (3) For For f >
1/2πτ1, Eq. (3.28) becomes SX(f) = 〈(∆X)2〉/π2τ1 ln(τ2/τ1)f2, which is rather rapid
decay to the power 2 in f . This attempt is successful in that it produces the spectra
of 1/f which is common in experiments. Equation (3.25) tells that the 1/f noise is
related to current flowing. However, experiments showed that even in equilibrium
situation without driving current or voltage, 1/f noise existed although the strict
thermal equilibrium may not be confirmed in experiments [81,82]
Hooge proposed an empirical expression of the 1/f noise for homogeneous samples
as
SI(f) = γI
2+β
Ncfα
SV (f) = γV
2+β
Ncfα,
(3.29)
with constants α, β, γ and the total number of charge carriers in the system, Nc [79].
He found that the value of γ is, surprisingly, constant about 2×10−3 for independent
electrons in homogeneous samples.
Shot Noise
The electrical conduction in circuit components is typically by majorities of electrons
or holes. Due to the nature of discreteness of electrons or holes, current becomes
50 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
statistically fluctuating around the mean value. The associated noise is called shot
noise. It results from the emission of discrete electrons. When the distribution of
electron emission follows the Poisson statistics, in other words, emitted electrons are
independent of each other, the shot noise is proportional to the average current value
I, i.e.
SI(ω) = 2eI, (3.30)
where e is the elementary charge. This noise is strong Particularly, this formula
is denoted as ‘full shot noise’. Unlike thermal noise, it emerges in non-equilibrium
situation, meaning that either voltage or current applies to the system. Shot noise
is also flat in range of frequency, satisfying fτ1 ≪ 1 where τ1 is the transit time of
electrons, supposedly very short. It is present in vacuum tubes and semiconductors,
where electrons leaves a cathode or passes a potential barriers.
Partition Noise
Ballistic conductors have no fluctuations in principle due to definite conductance
values. However, scatterers along the path of electrons or charge carriers contribute
a new source of noise. It is called partition noise. The incoming wave of particles is
partially reflected and partially transmitted after facing a scatterer. The probability of
the transmission by the scatter is denoted as T . The process is binomial distribution,
yielding the (1-T) fluctuations. The current fluctuation spectral density becomes
reduced from the full shot noise by this factor: SI = 2eI(1 − T ). When T is much
smaller than 1, this form goes back to the full shot noise value. The characteristic
of such scattering is elastic by conserving energy and momentum so that phase of
electrons is preserved, i.e. phase-coherent picture.
Quantum Noise
Quantum noise is the frequency-dependent excess noise. It is proportional to the fre-
quency, becoming the dominant fluctuations at high frequencies. To describe quan-
tum noise, correlations among electrons originated from Coulomb interactions and the
Pauli exclusion principle should be considered similar to the shot noise. Moreover,
3.2. FUNDAMENTALS OF NOISE 51
quantum noise includes vacuum fluctuations. The finite frequency current spectral
density is written with energy-independent transmission probabilities Tn
SI(ν) =N∑
n
Tn(1 − Tn)2e2
h
[
(eV + hν) coth
(
eV + hν
2kBT
)
+ (eV − hν) coth
(
eV − hν
2kBT
)]
+N∑
n
T 2n
2e2
h
[
2hν coth
(
hν
2kBT
)]
.
(3.31)
It is reduced to a simple form SI = 2hνG in the limit where hν ≫ eV, kBT with
G = 2e2/h∑
Tn. Equation (3.31) is a complete and general form of the current
spectral density including shot noise, thermal noise and quantum noise.
3.2.4 Crossover of Noise Sources in Frequency Domain
Figure 3.3 presents the frequency dependent current fluctuations SI versus frequency
f . Three distinct regimes are identified: (1) Region I: low frequency where 1/f noise
is a dominant noise source; (2) Region II: intermediate frequency where white noise
sources both thermal and shot noise are the biggest; (3) Region III: high frequency
where quantum noise exceeds other sources.
Pink Noise versus White Noise
Consider a system whose current power spectral density is composed of three main
components under assumption that the frequency is not high enough where quantum
noise does not play a dominant role: 1/f noise, thermal noise and shot noise. Assume
that all three are uncorrelated, then the overall noise power spectral density is a linear
combination such that
SI(ω = 0) = S1/f + SThermal + SShot
= AI2DC
f+ 4kBTG+ 2eIDC .
(3.32)
52 CHAPTER 3. MESOSCOPIC ELECTRON TRANSPORT
This equation produces the crossover from pink noise to white noise, which corre-
sponds to the left part of the plot in Fig. 3.3. In mesoscopic phase-coherent conduc-
tors, the excess noise is partition noise, so that the total noise power spectral density
at zero frequency is expressed with the transmission probability Tn
SI(ω = 0) = AI2DC
f+ 4kBTG
∑
n
Tn
+ 2kBTG∑
n
Tn(1 − Tn)
[
eV
2kBTcoth
eV
2kBT− 1
]
.
(3.33)
Note that there exists another crossover between thermal and shot noise. From
the second and third term in Eq.(3.33), thermal noise is bigger than the partition
noise as kBT ≫ eV . On the other hand, taking the limit of eV ≫ kBT , the last two
terms in Eq.(3.33) are reduced to∑
n(4kBTGQTn) + 2eI∑
Tn(1 − Tn)/∑
Tn. As T
goes to 0, the term equals to the shot noise value.
White Noise versus Quantum Noise
The second crossover occurs between white noise and quantum noise. The appropriate
formula is given by Eq.(3.31). There are several limiting cases in Eq.(3.31). First,
when eV ≫ kBT , Eq.(3.31) becomes SI = 2eI∑
Tn(1 − Tn)/∑
Tn, partition noise.
Second, when kBT ≫ eV , Eq.(3.31) recovers the Nyquist-Johnson noise such that
SI = 4kBTG. Last, when hν ≫ eV, kBT , the current spectral density is SI = 2hνG.
Therefore, the right part of Fig. 3.3 exhibits this crossover from shot and thermal
noise to quantum noise.
3.3 Summary
In this chapter, the basic theoretical descriptions of transport processes and transport
properties. The LRT as a fundamental theory to treat transport processes has been
reviewed and the Landauer-Buttiker formalism within the LRT has been introduced.
Based on this formalism, conductance and the current fluctuations are introduced.
The fluctuations and dissipation theorem connects non-equilibrium quantity with
3.3. SUMMARY 53
equilibrium quantity such that the shot noise is computed by the average value of
current. Various sources of current fluctuations have been discussed: thermal noise,
1/f noise, shot noise and quantum noise. The crossovers between different noise
sources in the frequency range have been examined.
Next chapter will discuss how to design practical experimental setup to measure
conductance and the current fluctuations. It will also present various techniques for
an improved signal to noise ratio.
Chapter 4
Experiment Methodology
Creative scientists have faith that
well-thought-out hypotheses, good experimental design, and persistence
will lead to truth through research.
− Robert V. Smith
Physics is a subject to explore natural and engineering phenomena with seeking
for fundamental understandings and to stimulate a search to discover intrinsic prop-
erties predicted by theoretical models. In all cases, both experiments and theories are
so tightly interconnected that theories without being confirmed by experiments and
experiments without being understood by theory are just incomplete. As an experi-
mentalist, I believe that the knowledge of physics is strongly rooted in empirical facts
and I emphasize the crucial roles of experiments not only as verification of existent the-
oretical speculations and hypotheses but also as a driving force to investigate further
unresolved phenomena beyond established models. Therefore, experiments should
be pursued for accurate measurements with a careful implementation of appropriate
apparatus. This chapter begins with a general guideline for electron transport mea-
surements with mesoscopic conductors whose specific outcome ensues in subsequent
chapters, and the guideline applies the hardware preparation to the specific case of
carbon nanotubes with specific numbers.
54
4.1. CONDUCTANCE 55
Of interest is to investigate electrical properties via conductance and shot noise
measurements in this thesis. For any experiments before actual implementation, the
initial step is to check the plausibility by rough estimate. It includes the recognition
of system limiting factors. As an example, conventional resistors cannot hold above
a certain voltage indicated by a power-rating: a 1/2-Watt 50 Ω carbon-composition
resistor can be biased up to 5 V before frying it. Similarly, mesoscopic conductors have
even tighter limit since they are too tiny to hold high voltage or current through them.
External excitation range would be constrained by the comparison of relevant energy
scalings. In order to observe the intrinsic properties in non-equilibrium states, the
bias energy to a system is greater than the thermal energy from non-zero temperature
of the ambience environment.
Next, the signal-to-noise ratio (SNR) of a system, one of the essential number
estimate, should be concerned. The SNR tells whether the signal is big enough to
be measured in the presence of possible noise sources. Plus, it provides how much
the signal should be increased for the bigger SNR with available techniques including
amplification or phase-sensitive detections. The SNR is defined in decibels (dB),
SNR = 10 log10
(
v2s
v2n
)
, (4.1)
where vs and vn denote a root-mean-square (rms) signal and a rms noise respectively.
It is unavoidable to handle weak signals from mesoscopic conductors, thus the main
efforts is devoted to improve the SNR. Here, two-terminal transport measurements
are focused.
4.1 Conductance
Mesoscopic field tends to report conductance rather than resistance from electron
transport measurements although conductance and resistance are reciprocal in gen-
eral. The conventional tradition is adopted here. The preceding chapter mentions
the distinction between linear conductance G and differential conductance dG. For a
linear current-voltage (I − V ) relation, two quantities are exactly same; however, the
56 CHAPTER 4. EXPERIMENT METHODOLOGY
nonlinear I − V trend occurs in most cases of mesoscopic systems, in which G and
dG become distinct. Since G is the integral of dG or dG is the differentiation of G
in mathematical perspective, one quantity can be computed from the other by post-
analysis. Instead of relying on numerical process, both are intended to be measured
experimentally. The method to extract a G value is straightforward such as applying
a dc voltage and measuring a dc current, whereas the method to find a dG value
requires some thoughts. The dG value corresponds to the tangential component at
a specific dc voltage or current, the bias components consist of both dc and ac such
that a tiny ac wiggle on top of a dc value. The ac change represents the slope at the
dc value in a I − V characteristic.
rIN
r1
r2
Rd
rINr1
r2
Rd
rIN
(a) (b)rIN
r1
r2
Rd
rIN
r1
r2
Rd
rINr1
r2
Rd
rIN
rINr1
r2
Rd
rIN
(a) (b)
Figure 4.1: (a) Two-terminal and (b) four-terminal measurement schematics with aconstant current bias.
Conductance is the response to the external stimulations acting on a system. A
couple of excitation schemes exist : current-bias and voltage-bias. It is not an ab-
solute rule to choose one scheme from the other, but a common sense for the best
results has been formed according to the size of conductance, such as a system whose
G is less than a quantum unit of conductance GQ (i.e. a high resistive conductor)
had better be biased by constant voltage source, whereas a highly conducting sys-
tem is biased by constant current source. A rationale behind the practical setup is
based on the stability of the bias source and the high SNR. In any bias schemes,
series resistance is unavoidable in a practical setup, originated from various sources
4.1. CONDUCTANCE 57
like wirings between equipments and contact interface between electrodes and de-
vice materials. Note that this issue is discussed readily in terms of resistance, thus
the following argument goes on with resistance not conductance. A typical two-
terminal measurement shown in Fig. 4.1 includes all spurious series lead resistances
ri together with a device resistance Rd. On the other hand, a four-terminal configu-
ration can eliminate series resistances, enabling us to access selectively an interesting
portion of circuits. Quantitatively, V = I(Rd + r1 + r2) in the former case and
V = V1 − V2 = I(Rd + r2)− Ir2 = IRd in the latter configuration under the assump-
tion that the output impedance of current sources and the input impedance rIN of
voltmeters are infinite.
Based on the actual implementation of transport measurements with single-walled
carbon nanotubes and quantum point contacts in the upcoming chapters, the setups
have several wiring options: coax cables, low-resistive wires and high-resistive wires.
We use a BeCu center conductor coax cables, whose resistance per a meter is 0.4 - 1
Ω, and a Cu wire has a low resistance around 3 Ω per meter and a Mn wire is used
as a high resistive wire, which has resistance around 200 Ω per meter. Coax cables
are necessary to protect signals from noise due to grounding of outer conductors,
low-resistive wires are preferable to bias electronic components, and high-resistive
wires are used to apply gate voltages where the fluctuations may be filtered out by
effective low-pass filter formed by resistance and capacitance. Since the resistance of
mesoscopic device is typically a range of 10 - 100 kΩ, the lead resistance of wirings
contributes negligibly.
The contact resistance, other source of series resistance, becomes an issue since
it may not be controllable easily. For a two-terminal single-walled carbon nanotube
(SWNT) device, the ideal resistance with a perfect Ohmic contact is 6.5 kΩ. The
resistance bigger than the ideal value indicates that the contact between the metal
electrode and the tube is just imperfect. Depending on how good the contact is,
the electrical transport regimes of the nanotube devices are classified as a tunnelling
regime with a strong barrier, namely poor contacts or a ballistic regime with a weak
barrier, namely good contacts. For SWNT devices, the physical contact cannot be
be separated from the intrinsic tube device since it is determined during fabrication;
58 CHAPTER 4. EXPERIMENT METHODOLOGY
thus, a four- terminal configuration does not win over a two-terminal counterpart. On
the other hand, the resistance of quantum point contact (QPC) can be extracted well
from all spurious resistance factors by the four-probe measurement in the Hall bar
geometry. In this reasoning, we performed a two-terminal measurement for SWNTs
and a four-terminal measurement for QPCs.
Suppose an experiment to measure a resistance around (10 kΩ)−1 with a 1 nA
current-bias. The expected voltage value according to Ohm’s law is around 1 µV,
which is too small to be detected in regular voltmeters. Due to engineering ad-
vancement, measuring 10 µV is not even a challenge any more by using commer-
cially available high-sensitivity multimeters or placing a low noise preamplifier af-
ter a device. The limiting factors to ruin an accuracy are noise sources. Whereas
prevalent extrinsic noise sources are eliminated by proper grounding and shielding of
noise reduction techniques [76], intrinsic noise sources cannot be completely excluded
so that strategic approach should be made. Amongst unavoidable intrinsic noise,
Johnson-Nyquist thermal noise is concerned, as an example, in a finite resistance R,
vthermal =√
4kBTRB where vthermal is a rms thermal noise in a unit of Volts, kB
Boltzmann constant, T temperature, and B the equivalent noise bandwidth. With a
100 kHz-bandwidth voltmeter at room temperature, vthermal ∼ 4µV, corresponding
to 40 % error and SNR = 20 log10 (10µV/4µV) ∼ 8 dB. The situation for the dG
case is even worse since an ac signal is supposed to be at least ten or hundred times
smaller than the dc value, thus the thermal noise in a wide frequency range exceeds
completely over the interesting signal, yielding SNR = 20 log10 (0.1µV/4µV) ∼ −32
dB. Given a R value, SNR enhancement can be achieved by decreasing T (cooling)
and B (narrow-band detection).
4.1. CONDUCTANCE 59
(a)
(b)
10-18
10-17
10-16
10-15
10-14
10-13
en
(V2/
Hz)
3 4 5 6 7 8
102 3 4 5 6 7 8
1002 3
T (K)
R = 10 kΩ
R = 100 kΩ
R = 1 MΩ
R = 10 MΩ
10-18
10-17
10-16
10-15
10-14
en
(V2/
Hz)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
T = 293 K
T = 77 K
T = 4 K
T = 1 K
(a)
(b)
10-18
10-17
10-16
10-15
10-14
10-13
en
(V2/
Hz)
3 4 5 6 7 8
102 3 4 5 6 7 8
1002 3
T (K)
R = 10 kΩ
R = 100 kΩ
R = 1 MΩ
R = 10 MΩ
10-18
10-17
10-16
10-15
10-14
en
(V2/
Hz)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
T = 293 K
T = 77 K
T = 4 K
T = 1 K
Figure 4.2: (a) Johnson-Nyquist noise vs. temperature for different resistance. (b)Johnson-Nyquist noise of resistors at various temperatures: room temperature (293K), liquid nitrogen temperature (77 K), liquid helium temperature (4 K) and 1 K.
60 CHAPTER 4. EXPERIMENT METHODOLOGY
(a)
(b)
Figure 4.3: (a) The 4 K home-made dipper and (b) Oxford Helium 3 sorption cryostat
4.1. CONDUCTANCE 61
Cooling Technique
Cooling is essential not only for a sake of thermal noise reduction but also to attain
degenerate electrons in mesoscopic conductors which is deeply related to the Fermi-
Dirac distribution at a given temperature. Figure 4.2 demonstrates the former role
of cooling that Johnson-Nyquist noise en = 4kBTR (V2/Hz) is dramatically reduced
at low temperatures. It is straightforward to cool down at 77 K and 4 K by simply
putting the system into appropriate cryogen, liquid nitrogen and liquid helium 4
(4He). Further temperature drop requires a well-thought design of cryostat. Pumping4He in a finite volume V reduces temperature down to ∼ 1.5 K, which is expected from
the ideal gas law PV = nRgT , the relation of pressure P and V and T with universal
gas constant Rg for n moles of ideal gas. Below 1.5 K, the isotope of helium 4, 3He
with a lighter mass due to one neutron deficiency is introduced. Condensation of3He gas and succeeding vapor pressure reduction bring the system temperature down
to 300 mK. As even lower temperature is required in transport measurements, the
dilution refrigerator is utilized, whose operation principle is based on natural phase
separation of the mixture of 3He and 4He below 700 mK [83]. The base temperature in
the dilution refrigerator remains continuously around 10 - 50 mK. The aforementioned
cryostats are commercially available even with magnet inside. The measurements
presented in succeeding chapters were done in a home-made 4 K dipper and a Helium
3 cryostat (Fig. 4.3).
Lock-in Detection
Lock-in amplifier often prevails over low-noise voltage or current preamplifier for a
weak signal. Its benefit lies on the concept of phase-sensitive detection [75]. Basically,
a signal is modulated at low frequency less than 100 Hz and a output signal at this
particular frequency is only captured by comparison to the same frequency reference.
In this way, the bandwidth as narrow as 10 mHz can be realized. All experimental
data in this dissertation were taken by 22 Hz modulation with SR830 lock-in amplifiers
and SR554 transformer preamplifier. The transformer preamplifier helps to reduce
a lock-in amplifier’s input noise with a gain of 100 or 500. This method applies
62 CHAPTER 4. EXPERIMENT METHODOLOGY
10-8
10-7
10-6
10-5
10-4
10-3
V
(V
)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
dc Voltage
ac Voltage
BW = 100 kHz BW = 1 Hz
(a)
(b)
10-7
10-6
10-5
10-4
10-3
V
(V
)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
dc Voltage
ac Voltage T = 293 K T = 4 K
10-8
10-7
10-6
10-5
10-4
10-3
V
(V
)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
dc Voltage
ac Voltage
BW = 100 kHz BW = 1 Hz
(a)
(b)
10-7
10-6
10-5
10-4
10-3
V
(V
)
104
2 3 4 5 6 7 8
105
2 3 4 5 6 7 8
106
R (Ω)
dc Voltage
ac Voltage T = 293 K T = 4 K
Figure 4.4: (a) Estimated dc (dot) and ac (square) voltages at a 1 nA current biasto a variable resistor together with Johnson-Nyquist noise measured by a 100 kHzequipment at room temperature and 4 K. Assume ac signal is hundred times smallerthan the dc value. (b) Johnson-Nyquist noise with a 1 Hz bandwidth at 4 K.
4.2. LOW-FREQUENCY SHOT NOISE 63
to any intrinsic noise reduction since the noise contribution is proportional to the
measurement bandwidth.
Combination of two methods boosts the ac signal up above the noise by two orders
of magnitude shown in Fig. 4.4 , enabling us to obtain the signal reliably. SNR for
both conductance and differential conductance is greatly improved by ∼ 80 dB and
∼ 40 dB respectively for a 10 kΩ resistor with a 1 Hz bandwidth detection at 4 K.
4.2 Low-frequency Shot Noise
The rationale described in conductance section applies to shot noise measurement
setup as well. As the first step, the signal and the noise are identified and SNR is
estimated accordingly whether present strategies are sufficient for accurate measure-
ment outcome. The signal carrying out useful information in this case is shot noise
and the remaining noise sources arise from thermal fluctuations and the amplifier
noise, assuming the extraneous noise sources are already well taken care of by the
noise reduction techniques.
For a maximum value of possible SNR, suppose the noise portion is small and the
signal is large such that the amplifier noise is zero and the value of current fluctuations
is the full shot noise since the mesoscopic conductor’s shot noise is suppressed due to
correlations among charge carriers. Then Eq. (4.1) in this extreme case is
SNR = 20 log10
( √2eIBR√
4kBTRB
)
.
Putting into some real values, consider that R equals to 20 kΩ and an average current
I is around 1 nA at 4 K. The estimated SNR is 20 log10(3.57 · 10−10/2.1 · 10−9) ∼ −15
dB. Since two noise sources are sharing the same bandwidth, there is no significant
gain by reducing the bandwidth. Certainly, cooling helps to enhance SNR; however,
the above value is at the best case. In other words, if we include the amplifier noise
and the suppressed shot noise, SNR is even lower than -15 dB. Therefore, other
strategic movements should be incorporated, which are AC modulation, a cryogenic
amplifier, resonant tank circuit design described below.
64 CHAPTER 4. EXPERIMENT METHODOLOGY
10-10
10-9
10-8
10-7
10-6
I (A)
103
104
105
106
107
R (
Ω)
Full shot noise dominant
Thermal noise dominant
10-21
10-20
10-19
10-18
10-17
10-16
10-15
SV
(V
2 /H
z)
10-8
2 3 4 5 6 7
10-7
2 3 4 5 6 7
10-6
I (A)
SI = 2eI, S
V= S
I(R)
2
SI(10k)2 SI(100k)
2
SI(1M)2 S
I(10M)
2
Sthermal = 10 kΩ
Sthermal = 100 kΩ
Sthermal = 1 MΩ
Sthermal = 10 MΩ @ 4 K(a)
(b)
10-10
10-9
10-8
10-7
10-6
I (A)
103
104
105
106
107
R (
Ω)
Full shot noise dominant
Thermal noise dominant
10-21
10-20
10-19
10-18
10-17
10-16
10-15
SV
(V
2 /H
z)
10-8
2 3 4 5 6 7
10-7
2 3 4 5 6 7
10-6
I (A)
SI = 2eI, S
V= S
I(R)
2
SI(10k)2 SI(100k)
2
SI(1M)2 S
I(10M)
2
Sthermal = 10 kΩ
Sthermal = 100 kΩ
Sthermal = 1 MΩ
Sthermal = 10 MΩ @ 4 K(a)
(b)
Figure 4.5: (a) Shot Noise and Thermal Noise crossover. (b) The threshhold R andI.
4.2. LOW-FREQUENCY SHOT NOISE 65
AC Modulation
in
V1
Rd
en
Lock
In( ) 2
V2 V3 V4
V
(a)
10
8
6
4
2
0
V (
mV
)
0.140.120.100.080.060.040.020.00
time (s)
(b)
in
V1
Rd
en
Lock
In( ) 2Lock
In( ) 2
V2 V3 V4
V
(a)
10
8
6
4
2
0
V (
mV
)
0.140.120.100.080.060.040.020.00
time (s)
(b)
Figure 4.6: (a) The circuit diagram of AC modulation scheme. (b) The square-wavevoltage in time.
Since the shot noise exists only when the average current flows (non-equilibrium
condition), AC full modulation of voltage bias, in principle, gets rid of unmodulated
noise, for example, thermal noise or amplifier noise together with the lock-in tech-
nique. Consider a simple conductor which has current fluctuations in (shot noise)
and voltage fluctuations en from thermal noise and amplifier noise shown in Fig. 4.6
66 CHAPTER 4. EXPERIMENT METHODOLOGY
(a). The voltage applying to the conductor to generate in is a square wave in time
domain at ωm (Fig. 4.6 (b)), and 100% modulation brings the offset to V = 0. Thus,
the bias voltage pattern indicates the conductor is biased on and off at frequency ωm,
resulting in a on-off in at the same frequency. Mathematically, a square wave consists
of harmonics of sine waves, thus time-dependent AC full modulation voltage bias and
in are written as,
V (t) =Vpp
2+Vpp
2
4
π
( ∞∑
n=1
1
(2n− 1)sin((2n− 1)ωmt)
)
=Vpp
2
(
1 +4
πsin(ωmt) +
4
3πsin(3ωmt) + ...
)
,
(4.2)
in(t) = in(0)
(
1 +4
πsin(ωmt) +
4
3πsin(3ωmt) + ...
)
, (4.3)
where Vpp is a peak-to-peak voltage value and in(0) is the device shot noise; for
example, in(0) =√
2eI for the full shot noise.
Now we can compute the values of Vi at each node i in Fig. 4.6(a):
V1 = en + in(t) ·Rd,
V2 = Av(V1) = Av(en + in(t) ·Rd),
V3 = V2 ·√B,
V4 = V 23 = A2
v(en + in(t) ·Rd))2B. (4.4)
Note that the dimension of V1 and V2 are [V/√
Hz] since en and in(t) are the square
of the noise power spectral density, but V3 after the band pass filter becomes [V] with
noise-equivalent bandwidth B. Av is the voltage gain of the amplifier. V4 is the value
proportional to the square of V3 in the unit of [V2]. The final readout of V at the
lock-in amplifier is from the demodulation of lock-in reference signal sin(ωmt). What
the demodulation at the lock-in does is to bring the component at sin(ωmt) to the
DC. Therefore, the surviving term in the end is the sin(ωmt) part. Plugging Eq. (4.3)
4.2. LOW-FREQUENCY SHOT NOISE 67
into Eq. (4.4) and assuming en and in are not correlated, V4 becomes
V4 = A2vB(e2n + (in(t) ·Rd)
2)
= A2vB[(e2
n + (in(0) ·Rd)2(1 +
(
4
πsin(ωmt)
)2
+
(
4
3πsin(3ωmt)
)2
+ 2
(
4
πsin(ωmt)
)
+ 2
(
4
3πsin(3ωmt)
)
+ 2
(
4
πsin(ωmt)
)(
4
3πsin(3ωmt)
)
+ ...)].
(4.5)
The relevant term for the demodulation is
A2vB(in(0) ·Rd)
22
(
4
πsin(ωmt)
)
,
which contains only the shot noise current fluctuations. Recall that sin2(ωmt) =1−cos(2ωmt)
2, the voltage at the lock-in is
V = A2vB(in(0) ·Rd)
22
(
4
πsin(ωmt)
)
sin(ωmt)
= A2vB(in(0) ·Rd)
22
(
4
π
1 − cos(2ωmt)
2
)
,
(4.6)
and the value of V measured is the dc component, Vout = A2vB(in(0) ·Rd)
2( 4π).
Cryogenic Amplifier
The benefits of placing amplifier close to the device are mainly two-fold: (1) to reduce
contamination to the signal from the device along the wirings between the device and
the amplifier. (2)to reduce the thermal noise of a transistor in the amplifier. We
used metal-semiconductor field effect transistors (MESFET) constructed in GaAs,
whose charge carriers are not frozen at low temperatures unlike Si. Plus, MESFETs
are advantageous due to a couple of features: high mobility of the carriers and high
transit frequency. The performance of a transistor depends on the bias points of
drain-source voltage VDS and gate-source voltage VGS as the source is referenced as a
ground. Figure 4.7 presents the IDS versus VDS at different VGS. The IDS−VDS trend
68 CHAPTER 4. EXPERIMENT METHODOLOGY
30x10-3
20
10
0
I DS (A
)
543210 VDS (V)
SONY MESFET
40x10-3
30
20
10
0
ID
S (A
)
2.52.01.51.00.50.0
VDS (V)
Fujitsu MESFET
(a)
(b)
30x10-3
20
10
0
I DS (A
)
543210 VDS (V)
SONY MESFET
40x10-3
30
20
10
0
ID
S (A
)
2.52.01.51.00.50.0
VDS (V)
Fujitsu MESFET
(a)
(b)
Figure 4.7: (a) SONY and (b) Fujitsu FSU01LG MESFET bias response at roomtemperature.
4.2. LOW-FREQUENCY SHOT NOISE 69
can be divided into three sections: Section I is where IDS increases as VDS increases
below 1 V; Section II is a upper right part in which IDS becomes constant for lower
absolute values of VGS; Section III is where IDS saturates around larger negative values
of VGS. In both Section II and III, the transistors have high gain, but they have high
power in Section II and low power in Section III. Meanwhile, when the transistors are
biased in Section I, they have reasonable gain but low power and low noise. Although
we cannot take advantage of high gain in Section I, achieved low power and low noise
are a good trade-off for a stable transistor performance.
Resonant Tank Circuit
In the shot noise measurements, there are several energy scales: thermal energy kBT ,
bias energy eV and measurement energy ~ω, which are relevant to thermal noise, shot
noise and quantum noise. In order to measure the shot noise, the following condition
should be satisfied, ~ω < kBT < eV . The lower bound of the measurement frequency
choice is limited by a 1/f noise corner frequency. A tank circuit is inserted after a
device to choose the measurement frequency at the resonant frequency of the circuit
ω = 1√LC
with a inductance L and a capacitance C. A simple tank circuit has two
degrees of freedom, resonant frequency ω and a quality factor (Q). We used the tapped
inductor tank circuit in order to introduce additional degree of freedom, impedance
matching because the optimal input impedance of a transistor is around 1 - 2 kΩ
lower than the device impedance ∼ 10 - 100 kΩ. A nice thing of this tank circuit
design is that it absorbs parasitic capacitance from coax cables, chip sockets and pc
boards and capacitance value change at different temperatures, shifting a resonant
frequency accordingly. We selected the adequate values of passive components for an
aimed frequency at low temperature around 10 - 20 MHz, which is high enough where
1/f noise is highly suppressed and ~ω ∼ 40 - 80 µ eV much less than kBT ∼ 0.3 meV
at 4 K and eV ∼ 1 meV.
70 CHAPTER 4. EXPERIMENT METHODOLOGY
Figure 4.8: The photograph of a cryogenic amplifier and resonant tank circuit usedin the single-walled carbon nanotube show noise experiments.
4.3 Summary
This chapter has discussed experimental schemes and techniques : measurement con-
figurations, bias schemes, cooling techniques, lock-in detection, AC modulation, cryo-
genic amplifiers, and a resonant tank circuit. These are practically used in actual
electrical measurements for conductance and shot noise properties towards the reli-
able and accurate measurement data. The outcomes and interpretations of applying
the techniques to specific conductors, single-walled carbon nanotubes and quantum
point contacts are discussed in two subsequent chapters.
Chapter 5
Single-Walled Carbon Nanotubes
We shall not cease from exploration.
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
− T. S. Eliot
The following two chapters present the direct application of the previously ac-
quired knowledge to actual systems : one class is single-walled carbon nanotubes and
the other is quantum point contacts in semiconductors. Both systems share common
features such that they are regarded as (quasi-) one-dimensional electron waveguides.
Chapter 5 is devoted to experimental and theoretical study of low-temperature trans-
port measurements through carbon nanotubes. The content starts off from an in-
troduction to single-walled carbon nanotubes, including band structure and device
fabrication and synthesis followed by experimental data of nanotube devices and the-
oretical interpretations of them.
71
72 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
5.1 Single-Walled Carbon Nanotubes
The discovery of single-walled carbon nanotubes (SWNTs) dating back to the be-
ginning of the 1990s has set out a new research trend widely spread over numerous
scientific and engineering fields. SWNT research themes evolve during the last two
decades, giving a clear hint that there is plenty of room to explore SWNTs in future.
The incessant interests on SWNTs arise from their distinct characteristics - in chem-
ical, mechanical, electrical, and optical aspects - and their potential contributions to
molecular electronics and optoelectronics. Almost defect-free surface, relatively easy
growth condition, and high geometric ratio of transverse dimension to longitudinal
length make SWNTs an ideal one-dimensional (1D) system. Therefore, SWNTs be-
come attractive study materials for low-dimensional physics in which strong electron-
electron interactions are unavoidable; and the innate tininess may allow SWNTs to
be a prospective replacement of silicon (Si)-based electronics.
It is a breakthrough in the SWNT research area that isolated individual SWNTs
were synthesized on Si-wafer with a reasonable yield by chemical vapor deposition
(CVD) in 1998 [15], enabling scientists and engineers to imagine limitless dream and
administer diverse tests. In addition, A part of reasons to achieve dense knowledge of
SWNTs in a concentrated period is the adaptation of Si-technology to SWNT-devices,
which has been well-established and optimized in a great extent. Indeed many en-
gineering components have been developed including sensitive chemical sensors [33],
intrajunction diodes [24], p-n junction [84], field-effect transistors [21], and single elec-
tron transistors [20, 22, 23]. Practically, various essential logic gates - AND/NAND,
OR/NOR, SRAM, inverter, ring oscillator - for memory storage and bit manipulation
for computation with superior performance over existent CMOS -electronics in terms
of low threshold and power consumption, implying a bright future [85,86].
Out of manifold approaches to assess characteristics, transport experiments have
been a straightforward and initial probe to extract thermal and electronic properties
arising from electrical and heat conduction by valence electrons and/or phonons in
device level once fabrication steps to place metal electrodes either on top of or under
5.1. SINGLE-WALLED CARBON NANOTUBES 73
carbon nanotubes were developed [19]. In particular, the electron transport measure-
ments have actively been performed all over the world last decade and have revealed
fascinating phenomena both in physics and in engineering aspects. In the contri-
butions to the fundamental physical knowledge, novel electronic properties through
SWNTs have been revealed such as Coulomb blockade oscillation [20], the Kondo
effect [25], ballistic quantum interference [16,17], and Tomonaga-Luttinger liquid be-
havior [23].
In this section, the basic knowledge about SWNTs nanotubes is introduced from
the bandstructure calculation and the different types of SWNTs are presented. And
the synthesis of SWNTs and the fabrication of SWNT devices are explained at the
end.
5.1.1 Electronic Band Structure
The carbon nanotube field was first initiated by identifying cylindrical objects ( now
called multi-walled carbon nanotubes ) in entangled carbon soots [2]. Immediately
theorists picked up the subject and moved rapidly to model this new system by sim-
plifying a one-shell tubule even before SWNTs were discovered in the lab yet. The
swift theoretical understanding of new materials was possibly obtained because their
mother material, the two-dimensional (2D) graphite or graphene, had been a long-
time subject in condensed matter physics. Regardless of having no idea how to form
such tubule things at lab benches, theorists envisioned that they would be shaped by
rolling one- or more than one- sheet of graphene seamlessly. Extra confinement in two
dimensions by this roll-up requires to satisfy a periodic boundary condition around
the circumferential direction and leaves room to have only one freely propagating di-
rection. The task became to apply the new boundary condition to the band theory of
graphene based on a tight-binding calculation of unpaired π-orbital [5]. This yielded
an appropriate band structure of SWNTs in 1992 by several groups [3, 4]. Following
the historical path of obtaining SWNT bandstructure, the beginning step is to study
graphene band structure.
74 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
a1
a2
O A B
a1
a2
O A B
x
y
x
y
r1
r2
r3
r1
r2
r3
kx
ky
kx
ky
K
K′
b1
b2
M
Γ
(a)
(b) (c)
Figure 5.1: (a) Graphite lattice structure. (b) The direct and (c) the reciprocal lattice
space of graphene with unit vectors ~ai, ~bi and translational vectors ~ri.
5.1. SINGLE-WALLED CARBON NANOTUBES 75
Graphene
Graphene is a name to indicate only one sheet of three dimensional graphite which
contains many layers of sp2 carbon hexagons shown in Figure 5.1(a). Considering
the fact that the spacing between layers (3.35 A) is longer than the neighbor atom
distance aC−C (1.42 A) within the same plane, thus two dimensional graphene can
exist in nature. Unlike the previous statement, it has been a challenging to isolate one
layer from bulk graphite. Recently several groups succeeded to construct graphene
reproducibly by mechanical exfoliation, which fires the second boom of graphene
research [34,35].
Figure 5.1 presents the direct and reciprocal lattice structures of graphene, show-
ing a hexagonal Brillouin Zone (BZ). For consistency, a horizontal axis in figures
always represents ~x and a vertical one does ~y. Two unorthogonal unit vectors ~a1 and
~a2 form a unit cell. The unit cell contains two carbon atoms A and B in the direct
lattice space. A single carbon atom couples to three nearest neighbor carbon atoms
and their corresponding translational vectors are denoted as ~ri where i goes 1, 2, and
3. Similarly, two unit vectors ~b1 and ~b2 are readily found from the direct lattice unit
vectors in 2D, satisfying ~bi · ~aj = 2πσij by definition. The (x, y) coordinates of these
unit vectors from the origin O = (0, 0) are specified with the size of the unit vectors
a = |~a1| = |~a2| =√
3ac−c,
~a1 = (a
√3
2, a
1
2),
~a2 = (a
√3
2,−a1
2),
~b1 = (2π√3a,2π
a),
~b2 = (2π√3a,−2π
a),
76 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
~r1 = (−a√
3
6, a
1
2),
~r2 = (−a√
3
6,−a1
2),
~r3 = (a1√3, 0).
In momentum k-space (Fig. 5.1 (c) ), the high symmetry points Γ (the center of the
first BZ), K, K ′ (vertices of a hexagon) and M (a middle point in the side of the
hexagon) are indicated. Note here that all vertices of the first BZ are inequivalent.
They are distinguished by either K or K ′. This inequivalence can be easily checked
by rotating each vertex by 2π/3. And the alternate order of K and K ′ are easily
assigned. The unit cell in the BZ contains two vertices K and K ′.
Let us restrict our focus on the unit cell and the first BZ. A Hamiltonian H is writ-
ten using the tight-binding calculation. Tight-binding theory is a good approximation
of atoms in well-defined lattice sites. It describes the effect on a localized electron in
an atom due to the existence of nearest neighbor entities perturbatively [60]. It is a
well-known fact that three valence electrons per each carbon atom form sp2 hybridiza-
tion. The chemical properties are described by the remaining unpaired π electron in
a 2 pz orbital. Thus, the Hamiltonian H can be expressed by a simple 2-by-2 matrix
for the unpaired electron in two carbon atoms in the unit cell. Obviously, the bases
are pz atomic orbitals of carbon atom A and B denoted as Φi(~k, ~r) where i = A and
B. These wavefunctions should satisfy Bloch’s theorem due to periodicity or transla-
tional symmetry i.e. Φi(~k, ~r) = Φi(~k, ~r+~a) where ~a is a translation vector. The form
of H is
H =
(
HAA HAB
HBA HBB
)
,
where HAA = HBB = ǫ2p and HAB = t∑
~riei~k·~ri . ǫ2p is the pz orbital energy and
it is often set to 0 for convenience. t is the transfer integral between two atoms A
and B. Conventionally, t is negative and ~ri is a nearest neighbor translation vector.
5.1. SINGLE-WALLED CARBON NANOTUBES 77
Therefore, the matrix elements are explicitly rewritten as
H =
(
ǫ2p tf(~k)
tf∗(~k) ǫ2p
)
where f(~k) = ei~k·~r1 + ei~k·~r2 + ei~k·~r3 .
The remaining tasks are to obtain eigenvectors and eigenvalues by solving a secular
equation, det(H−EI) = 0 where I is a 2-by-2 identity matrix. The two eigenenergies,
E1 and E2 are ~k-dependent. The corresponding eigenfunctions Ψ1 and Ψ2 are a linear
combination of the two bases vectors:
Ψ1 ≡(
CA1
CB1
)
= CA1ΦA(~k) + CB1ΦB(~k),
Ψ2 ≡(
CA2
CB2
)
= CA2ΦA(~k) + CB2ΦB(~k).
Chemists have special terminologies for orbital bonding. When CA/CB = +1,
two pz orbitals face each other in the same direction, technically called ‘π-bonding’
whereas the negative -1 state forms ‘π∗-bonding (anti-π bonding)’
Generally the overlap matrix S of orbitals is not zero and needed to be considered
as well since the two wavefunctions are not completely isolated from each other,
S =
(
1 sf(~k)
sf ∗(~k) 1
)
where s is the overlap integral between two Bloch wavefunctions. In the Slater-Koster
scheme, s is set to 0 for a simple approximation of the band structure calculation in
graphite.
When both ǫ2p and s are assumed to be zero, eigenvalues are
Ei(~k) = ±t√
∣
∣
∣f(~k)
∣
∣
∣
2
78 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
where i = 1 and 2. The interesting feature in Ei(~k) versus ~k is that a gapless disper-
sion at vertices of the first BZ occurs. That is why graphene is semi-metals shown in
Fig. 5.2. Also note that near EF , the dispersion is linear so that it becomes massless
Dirac dispersion. The related phenomena due to massless Dirac dispersion in the
quantum Hall regime have been reported with graphene recently [34,35].
Figure 5.2: Energy band structure of Graphene.
Single-Walled Carbon Nanotubes
Figure 5.3 describes an imaginative way to form SWNTs by placing a line OB over
a line AB′. There are infinite ways to roll up the graphene sheet in principle. A
specific SWNT among many possible configuration is identified by two orthogonal
vectors, chiral vector ~Ch and the translation vector ~T . Two vectors are linked to the
graphene unit vectors as a linear combination with positive integers n and m such as
~Ch = n~a1 +m~a2. Since hexagonal symmetry sets that in the regime of 0 < |m| < n,
all possible SWNTs configurations appear. Therefore, a group of coefficients (n,m)
classifies SWNTs. This notation is very convenient: once (n,m) is determined, all the
relevant parameters can be readily computed, for example, ~T , the chiral angle θ, the
5.1. SINGLE-WALLED CARBON NANOTUBES 79
number N of hexagons per unit cell , the diameter of a tube dt, and corresponding
reciprocal vectors ~K1, ~K2.
Figure 5.3: SWNT geometry on the graphene lattice structure. Chiral vector ~Ch isdrawn for a specific case, a (4,2)-SWNT.
A periodic boundary condition along the chiral vector or ~K1 quantizes reciprocal
vectors in infinitely long SWNTs so that the true reciprocal lattice vectors are in the
direction of ~K2. The reciprocal vectors are, therefore, represented as k ~K2/∣
∣
∣
~K2
∣
∣
∣+µ ~K1
where k is a continuous-variable wavenumber in the longitudinal direction. Since the
number of reciprocal vectors is N , the reciprocal vector index µ takes an integer value
from 0 to N − 1 in a unit cell.
General expressions of the aforementioned parameters for a (n,m) tube are sum-
marized as follows:
• ~Ch = n~a1 +m~a2 ≡ (n,m) where 0 < |m| < n,
• dR = gcd(2n+m, 2m+ n) where ‘gcd’ means the greatest common divisor,
• ~T = t1 ~a1 + t2 ~a2 where t1 = n+2mdR
and t2 = −2n+mdR
,
• cos θ =~Ch· ~a1
| ~Ch|| ~a1|= 2n+m
2√
n2+nm+m2
• N =| ~Ch×~T || ~a1× ~a2| = 2(n2+nm+m2)
dR
80 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
• dt =| ~Ch|
π= a
π
√n2 + nm+m2
• ~K1 = 1N
(−t2~b1 + t1~b2),
• ~K2 = 1N
(m~b1 − n~b2)
•∣
∣
∣
~K2
∣
∣
∣= 2π
|~T |
Inserting the relations of ~K1 and ~K2 into the E(~k) obtained in the above, the band
structure of SWNTs are analytically calculated. Notice that according to where the
one-dimensional BZ of SWNTs meets the graphene BZ, SWNTs exhibit metallic or
semiconducting electronic properties. Two extreme cases are considered as examples:
Armchair (n,n) SWNTs for a truly metallic case and zigzag (n,0) SWNTs for either
metallic or semiconducting cases. The denotation of armchair and zigzag comes from
the shape of carbon hexagon array along the SWNT surface.
Armchair((n,n)) Single-Walled Carbon Nanotubes
Armchair SWNTs are special in a sense that ~Ch is in x direction from the same
coefficients of ~a1 and ~a2 due to n = m. Consequently, allowed reciprocal vectors are
parallel to y direction (Fig. 5.4). The 1st BZ has a length of 2π/a, independent of
index n. Note that the energy band of µ = n is the lowest state in which ~K2 passes
through inequivalent vertices of the hexagon. This µ value comes from the ratio of
two length scales |ΓM | and∣
∣
∣
~K1
∣
∣
∣.
The above parameters are reduced to have simple expressions:
• Ch = n~a1 + n~a2 ≡ (n, n),
• dR = 3n,
• T = t1 ~a1 + t2 ~a2 where t1 = 1 and t2 = −1,
• cos θ =√
32,
• N = 2n
5.1. SINGLE-WALLED CARBON NANOTUBES 81
a1
a2
O A
x
y
Ch
T K
K’
K2
K1kx
ky
b1
b2
(a) (b)
(c)
¡B
a1
a2
O A
x
y
x
y
Ch
T K
K’
K2
K1kx
ky
kx
ky
b1
b2
(a) (b)
(c)
¡¡B
Figure 5.4: Armchair SWNT real (a) and reciprocal (b) lattice space. (c) (10,10)energy band structure. The first BZ of SWNT is indicated by two thick vertical lines.
82 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
• dt = aπ
√3n
• ~K1 = 12n
(~b1 + ~b2)
• ~K2 = 12(~b1 − ~b2)
•∣
∣
∣
~K2
∣
∣
∣ = 2πa
The case n = 10 is computed as a specific example. SWNTs with the (10,10)
configuration are obtained dominantly in laser ablation synthesis according to the lit-
erature. The bandstructure clearly shows that there is no gap at the Fermi level since
the lowest energy band always crosses at the vertices within the 1st BZ. Therefore,
the finite density of states at the Fermi level confirm that armchair tubes are metallic.
In addition, the real and imaginary coefficients of eigenfunctions assign the π and π∗
bands without confusion. The crossing bands at the Fermi energy are orthogonal,
therefore, the interband scattering is blocked by symmetry. This explains well that
metallic tubes do not suffer from backscattering processes near kF and they have long
mean free paths [87].
Zigzag((n,0)) Single-Walled Carbon Nanotubes
Zigzag SWNTs have quantized reciprocal vectors placed with a spacing of∣
∣
∣
~K1
∣
∣
∣. They
are perpendicular to a line between Γ and K shown in Fig. 5.6. These vectors are
along a ~b2. Let us compare two length scales.
|ΓK|∣
∣
∣
~K1
∣
∣
∣
=
(
4π3a
)
(
2πan
) =2n
3. (5.1)
Equation (5.1) differentiates two groups of (n,0) SWNTs. As n is a multiple of 3,
a reciprocal vector overlaps with one of vertices in a hexagonal BZ. While another
group exists when the ratio is not an integer because n is not a multiple of 3. Then,
a reciprocal vector misses any of vertices K or K′, creating the energy band gap.
Depending on n values, zigzag tubes are either metallic (n = 3q where q is a positive
integer) or semiconducting. Semiconducting tubes have a direct band gap at a certain
5.1. SINGLE-WALLED CARBON NANOTUBES 83
K’K
Γ
π
*π
*π
π*π
π
(a)
(b)
(c)(c)
(c)
Figure 5.5: (a) The µ = 0 band for a (10,10) SWNT along symmetry points Γ, K, andK ′. The π, π∗ wavefunctions are clearly denoted based on the energy band coefficients.The real and imaginary coefficients of upper energy band (b) and lower energy band(c) for µ = 0. The dotted line is the zone boundary of graphene.
84 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
a1
a2
O A
x
y
T Ch K
K’
K2
K1
kx
ky
b1
b2
(a) (b)
(c)
B ¡
a1
a2
O A
x
y
T Ch K
K’
K2
K1
kx
ky
kx
ky
b1
b2
(a) (b)
(c)
B ¡¡
Figure 5.6: A zigzag SWNT unit cell in a real (a) and a reciprocal (b) lattice space.(c) Semiconducting zigzag (10,0) energy band structure. The first BZ of SWNTis indicated by two thick vertical lines. (d) µ = 6 band for (10,0). The real andimaginary coefficients of upper energy band (e) and lower energy band (f) for µ = 6.
5.1. SINGLE-WALLED CARBON NANOTUBES 85
(a)
(b)
(c)
(a)
(b)
(c)
Figure 5.7: (a) The µ = 6 band for a (10,0) zigzag SWNT. The real and imaginarycoefficients of upper energy band (b) and lower energy band (c) for the µ = 6 band.
86 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
wavenumber k. The gap energy is typically about 1 eV or so and its size is inversely
proportional to a diameter (d), Eg ∝ eV/d. The fact that they are direct bandgap
materials and the bandgap 1 eV is close to the communication optical wavelength
is advantageous for optoelectronic and photonic devices as engineering applications
in future. Figure 5.7 shows a particular case for n = 10, which corresponds to a
semiconducting zigzag tube. In this case, the assignment of π and π∗ is not clear
since the bands are actually mixed and expressed as a superposition of two bands
plotted in Fig. 5.8. This reason makes semiconducting SWNTS to have shorter mean
free paths due to backscattering processes.
This rather simple theory with a (n,m) notation has enabled us to gain remarkable
insights into material structures and these knowledge have been confirmed by scanning
tunnelling microscopy experiments [13,88].
5.1.2 Synthesis and Fabrication
Discovering multi-walled carbon nanotubes (MWNTs) seems fortuitous in a carbon
arc-discharge chamber which was designed to the production of fullerenes [2]. Two
years later, SWNTs were found by the arc-discharge method similar to MWNTs this
time except for adding catalytic components in the chamber [8]. Although these
works sparkled scientists and engineers’s interest significantly enough to establish
a huge community of theoretical and experimental research on nanotubes, efficient
synthesis methods have been on demand in order to isolate nanotubes, to grow specific
nanotubes, and to build up refined devices to investigate novel quantum phenomena
in one dimension for quantitative assessment. Out of three major approaches to
synthesize SWNTs using catalytic nanoparticles, electric arc-discharge, laser ablation
and chemical vapor deposition (CVD), the CVD method has been superior to produce
high-quality SWNTs.
Synthesis: Chemical Vapor Deposition
CVD refers to a chemical process to deposit a thin-film or dense structures like pow-
ders or fibers on substrates using gaseous reactants. The basic principle of operation is
5.1. SINGLE-WALLED CARBON NANOTUBES 87
to flow a gas phase of elements or compounds into a substrate, on which the supplied
gas will undergo thermal reaction and will be decomposed while reaction byprod-
ucts are flushed out. Its versatility of any element or compound, high-purity, high
density, low cost, simplicity and flexibility to variations explicate the wide-usage and
numerous forms of CVD in semiconductor industries. Diverse CVD systems share
four common structures: a reaction vessel, a source of reactants, a substrate, and an
exhaust system for byproduct removal.
The CVD chamber which was used to grow SWNT for devices presented in this
dissertation also consists of a similar structure ( Fig. 5.8 (a)): a 1-inch diameter tube
vessel inserted into a furnace, sources of gas CH4, H2 and Ar, a Si-substrate con-
taining catalyst islands and an exhaust system. Using iron based alumina-supported
catalyst, SWNTs were grown with a carbon feedstock, 99.999 % CH4 and H2 at right
concentration for 5 - 7 minutes at 900 - 1000 oC, followed by a Ar flush and a cool-
down to room temperature. The success to synthesize high-yield of SWNTs near
the catalyst islands with low resistance has advanced the SWNT research field into
ballistic transport studies and prototype of nanotube-electronics [15].
The synthesis mechanism in the catalytic CVD method is associated with the
details of nanoparticles since the catalytic nanoparticles are essential to form the
SWNTs unlike then MWNTs. Recently, Li et al. have attempted to assess the role
of catalysts. The authors have showed that the diameter of SWNTs indeed closely
linked to the nanoparticle size in terms of statistical analysis [89]. According to
this report, the synthesis can be understood in three stages: first, nanoparticles as
catalysts absorb decomposed carbon atoms from CH4 or other carbon feedstock in the
CVD process; second, the absorption of carbon atoms to nanoparticles would continue
until the saturation. Once it reaches the saturation, carbon atoms become to grow out
from the catalysts with a closed-end; third, an excess carbon supply adds to carbon
precipitation on surface and it yields finite-length nanotubes in the end. Figure 5.8
(b) illustrates such speculative synthesis mechanism by computer simulations from
Professor K. J. Cho group. It is reasonable, therefore, that the SWNT diameter
would be determined by the nanoparticle size as the initial basis. Although Li et al.
[89] provided valuable information as to the microscopic level understanding of the
88 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
CH4
H2
Ar
valve
Furnace
Exhaust
(a)
(b)
CH4
H2
Ar
valve
Furnace
Exhaust
CH4
H2
Ar
valve
Furnace
Exhaust
(a)
(b)
Figure 5.8: (a) Schematics of CVD chamber. (b) The mechanism of SWNT growthfrom catalysts in CVD chamber adapted from K. J. Cho group.
5.1. SINGLE-WALLED CARBON NANOTUBES 89
synthesis in the catalytic CVD process, the complete controllability to produce tailor-
made SWNTs with an expected diameter, chirality, length, position and orientation
at will is yet to be far from reality. This is the SWNT fabrication challenge at
present. Once this goal is achieved, it is not difficult to imagine that SWNTs would
become ubiquitous in various applications as electrical, chemical, mechanical and
optical components.
Fabrication Processes
The configuration of SWNT devices this thesis studies resembles conventional semi-
conductor field-effect transistors, which have three terminals: source, drain and gate.
The fabrication goal is to produce three-terminal isolated SWNT nanotube devices
on top of a Si-wafer.
Figure 5.9 shows the steps of processes. The procedure has been optimized to have
a high yield of SWNTs devices appropriate for experimental purpose. The starting
material is a four-inch heavily doped p-type Si-wafer. It is substrate which serves
as a backgate. Due to multi-layer lithographic steps, first, global wafer marks and
chip marks should be placed in the blank wafer. These marks are very useful firstly
that the overlapping processes can be performed within the lithographic resolution
limit and secondly that they draw boundaries of chips in the whole wafer, where total
64 chips are located. Later the wafer is broken into each chip along the alignment
marks for further steps. In general the sizes of marks are 2 µm - features which are
large enough to be recognized easily. Those marks are patterned by a standard 1
µm-photolithography recipe developed in the Stanford Nanofabrication Facility. The
wafer is coated by a 3612 positive photoresist for 1 µm-thick resist layer in a svgcoat
machine, and it is exposed for 30-40 seconds using a EV aligner or Karl Suss exposers.
After developing at a svgdev track, the wafer is loaded in a Drytek for etching. It
etches the Si-substrate by 1 µm in depth. Descum process before etching to get rid
of residual photoresist around trenches would be of use in order to produce sharp
edges. The photoresist layer is removed at wet-bench non-metal. The marks are
basically engraved in the Si-wafer, they are robust and chemically inactive for any
further steps.
90 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
Thermal oxidation
Si
SiO2
PMMA
Si
SiO2
E-beam for catalyst islands
SWNT synthesis
Si
SiO2
PMMA
E-beam for electrodes
Si
SiO2
Metal deposition and Lift-off
Alignment marks Thermal oxidation
Si
SiO2
PMMA
Si
SiO2
Si
SiO2
E-beam for catalyst islands
SWNT synthesis
Si
SiO2
PMMA
Si
SiO2
Si
SiO2
PMMA
E-beam for electrodes
Si
SiO2
Si
SiO2
Metal deposition and Lift-off
Alignment marks
Figure 5.9: The schematics of SWNT-device fabrication processes.
5.1. SINGLE-WALLED CARBON NANOTUBES 91
The second step is the thermal oxidation on top of the marked Si-wafer. It is
a very critical step to avoid any possible impurities on wafers, since any dirts on
the wafers will lead to a leakage when devices are characterized. Therefore, before
inserting the wafer to a diffusion furnace named Tylan for thermal oxidation, the
wafer should be cleaned thoroughly and properly through the diffusion wet bench
process. The 0.5 -µm oxide layer is grown on the wafer. Again up to this point,
an extreme care to prohibit contaminations should be taken. With the oxide-layer,
the four-inch wafer is now broken into pieces for further delicate processes as a chip.
A rough dimension of chip is about 4 mm × 4 mm. The next task is to pattern
catalyst islands at specific locations for growing SWNTs. The size of catalyst isalnds
are about 5 µm × 5 µm and they are defined relative to chip marks by electron-beam
lithography (EBL) using Hitachi H-700 F11 Electron Beam System. The Hitachi
system has a minimum feature size around 70 nm reproducibly. One of common EBL
resist is polymethylmethacrylate (PMMA). For this step, a 5 % 495 K PMMA is
coated. Alumina and iron based catalyst material solution is dropped on the surface
and the chip is inserted into an oven for 5 minute to dry out the catalyst solution.
The PMMA layer is removed in acetone.
Nanotubes are synthesized by the aforementioned CVD method with methane
and hydrogen gas at 900 ∼ 100 o C at the furnace for 5 ∼ 7 minutes. Once the
furnace is cooled down to room temperatures, the growth yield is roughly checked
by atomic force microscopy (AFM) in order to know the distribution of tube density
near the catalysts. Since an isolated SWNT is in question, the low number of SWNTs
grown near the catalyst island is perferrable. The temperature and the duration of
the growth are slightly modified each time according to the tube density analysis.
The second EBL is proceeded for patterning metal electrods after PMMA coating,
exposure and development. This is followed by the metal deposition in the Innotec or
Dai-group metal evaporator. The choices of metals have been Titanium (Ti) only, Ti
and gold (Au), and palladium (Pd) only. The typical thickness of metals are around
50 ∼ 70 nm. The liftoff in acetone to remove PMMA is executed as the final step.
The portion of some devices in one chip out of the fabrication steps is imaged by
optical microscopy shown in Fig. 5.10(a). The total 98 individual devices are made
92 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
in one chip and figure 5.10 presents the zoom-in image of an individual device. The
two big squares are the top and bottom 100 µm × 100 µm pads for wire-bondings
and the two black middle squares are the catalyst islands. Two narrow thin lines are
electrodes on top of the tube to be source and drain terminals.
Room Temperature Characterization
As a chip containing 98 devices is ready to be tested from the fabrication processes
described above, it undergoes a couple of characterization steps at room temperature
(RT), which facilitate to filter out well-contacted devices with an individual metallic
SWNT before a cool-down. First, the I − Vds at a few Vg is measured of each device
at a probe station. According to the resistance (R) and its dependence on Vg, devices
whose R is between 10 ∼ 30 kΩ and which have weak or no dependence on Vg (metallic
behavior) are selected for the atomic force microscopy (AFM) imaging. The AFM
images determines the number of nanotubes across the source and drain electrodes
and the diameter of SWNTS (Fig. 5.10(c)). Nanotubes, whose diameters are 1.5 ∼3.5 nm from AFM images, are presumably regarded as SWNTs based on statistics.
AFM images cannot identify a SWNT, a double-walled nanotube or multi-walled
nanotube at all unlike transmission electron microscopy (TEM) which requires a
conducting substrate. The current device configurations for transport measurements
are inadequate to TEM, thus the choice of SWNTs relies on statistics of TEM results
with synthesized nanotubes on conducting substrates by the same recipe.
5.2 Differential Conductance
According to the RT characterization results, chosen SWNT-devices were wirebonded
at a chip socket. They were loaded in the dipper for a cool-down at 4 K. Figure 5.10(a)
shows the optical microscopy picture of several wirebonded devices in a chip. The
SWNT devices have a three-terminal geometry: source, drain and backgate whose
diagram is simplified in Fig. 5.11 (a). The Si substrate was used as the backgate. The
dimension of metal electrodes was aimed as around 200 nm, and the spacing between
two electrodes was determined by the EBL pattern, between 200 and 600 nm. The
5.2. DIFFERENTIAL CONDUCTANCE 93
(a)
(b)
(c)
500nm
100¹m
100¹m
(a)
(b)
(c)
500nm500nm
100¹m100¹m100¹m
100¹m100¹m
Figure 5.10: (a) Optical microscopy picture of portion of a chip containing wire-bonded devices, (b) a zoom-in view of an individual device, and (c) atomic forcemicroscopy image of an individual SWNT with patterned Ti/Au electrodes.
94 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
Ti/Au, Ti-only, and Pd metal electrodes were used, which featured low-resistance
contacts. The resistance of the selected metallic SWNT devices was typically 12 ∼50 kΩ at room temperatures and about 9 ∼ 25 kΩ at 4 K. Note that the resistance
of tube devices decreases as the temperature decreases. The trend of resistance and
temperature can be explained by phonons, which are frozen at low temperatures. It
is distinct from devices which are well isolated from the electrodes, in that resistance
becomes higher at low temperatures.
5.2.1 Quantum Interference
SWNTs well-contacted to two electrodes with finite reflection coefficients produce
quantum interference pattern in differential conductance ( dI/dVds ) at 4 K. Predic-
tions [87, 90] and experiments [16, 17, 20, 21] have shown that both the elastic and
the inelastic mean free path are at least on the order of microns in metallic nan-
otubes at low temperatures. Therefore, the electron transport within 200 ∼ 600
nm-long SWNTs is believed to be ballistic [16, 17]. Figure 5.11 (b) is a represen-
tative two-dimensional image plot of dI/dVds as a function of a drain-source voltage
Vds in y-axis and a backgate voltage Vg in x-axis. The color bar scale is renormalized
by 2GQ = 4e2/h. The interference pattern arises from the fact that SWNT has a
finite length determined by the electrode spacing. The diamond structures at low
Vds are an additional confinement along the longitudinal direction due to the poten-
tial barriers at the interfaces with two metal electrodes. The confinement quantizes
energy levels and the energy spacing between maxima corresponds to ∆E = ~vF/L.
The Fermi-velocity of SWNT is adopted from the value of graphite Fermi-velocity,
8 × 105 m/s. The oscillations are measured from a 360 nm-long SWNT. The cor-
responding energy spacing is ∆E ∼ 10 meV. The size of the pattern is consistent
with the energy spacing in a good agreement with the experiment results. Liang et
al. reported similar interference features up to 5 mV Vds values with 200 nm and
500 nm-long Ohmic contacted SWNT devices at 4 K [16]. They modelled the system
as an electronic analog to the Fabry-Perot (FP) cavities, and they claimed that the
observation of quantum interference is an evidence of the ballistic transport. More
5.2. DIFFERENTIAL CONDUCTANCE 95
Gate
SiO2
g = 1 g = 1g < 1
DrainSource
Vg
Vds
-10 -9 -8 -7 -6 Vg (V)
-20
-10
0
10
20
Vds
(mV
)0.40
0.38
0.36
0.34
0.32
0.30
0.28
0.26
-9.0 -8.5 -8.0 -7.5 -7.0
-40
40
Vg (V)
(a)
(b)
Gate
SiO2
g = 1 g = 1g < 1
DrainSource
Vg
Vds
Gate
SiO2
g = 1 g = 1g < 1
DrainSource
Vg
Vds
-10 -9 -8 -7 -6 Vg (V)
-20
-10
0
10
20
Vds
(mV
)0.40
0.38
0.36
0.34
0.32
0.30
0.28
0.26
-9.0 -8.5 -8.0 -7.5 -7.0
-40
40
Vg (V)
-10 -9 -8 -7 -6 Vg (V)
-20
-10
0
10
20
Vds
(mV
)0.40
0.38
0.36
0.34
0.32
0.30
0.28
0.26
-9.0 -8.5 -8.0 -7.5 -7.0
-40
40
Vg (V)
(a)
(b)
Figure 5.11: (a) The schematics of three-terminal SWNT device. (b) Experimentaltwo-dimensional image plot of differential conductance versus a drain-source voltagein y-axis and a backgate voltage in x-axis.
96 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
specifically, the phase information preserves along the nanotube. Contrary to the
usual FP oscillations, we found in all devices that the interference pattern fringe con-
trast is reduced in magnitude at high Vds shown in Fig. 5.11 (b). In order to explain
experimental observations in dI/dVds, three theoretical modesl are investigated in
this section: A single-channel double-barrier structure model, two-channel double-
barrier model in the Landauer-Buttiker formalism and a double-barrier cavity in the
Tomonaga-Luttinger liquid theory.
Double-Barrier Structure
A simple theoretical attempt to study Vds-dependent differential conductance (dI/dVds)
for a finite-length device is to compute dI/dVds in a double-barrier structure with two
transmission coefficients TL and TR of left and right barriers. Assume that all four
channels from spin and orbital degeneracy are acting identically, thus a factor 4 ap-
pears in the expression of dI/dVds. Considering the multiple reflections between two
barriers, the overall transmission coefficient T is energy (E) dependent,
T (E) =TLTR
1 + (1 − TL)(1 − TR) − 2√
(1 − TL)(1 − TR) cos(φ(E)), (5.2)
where the accumulated phase from the reflection paths is φ(E) and it is a function
of Vds, Vg and the SWNT length L. From an empirical trace, the maximum and
minimum value of T can be extracted because the maximum of Eq.(5.1) occurs at
cos(φ) = 1 and the minimum does at cos(φ) = −1, respectively. The explicit expres-
sion of φ is φ = EL~vF +αVg = eVdsL/~vF +αVg. The constant α is the effectiveness
of the gate voltage to the device through the oxide layer. In our case, α is around
0.01 from the quantum dot devices with the same thickness of a SiO2 oxide layer.
From the Landauer-Buttiker formula, dI/dVds is given by
dI
dVds
=4e2
hT (E) =
4e2
hT (Vds, Vg). (5.3)
The theoretical fitting to the experimental data is done in Fig. 5.12. This simple
model fits well the experimental dI/dVds data at low Vds, but it deviates significantly
5.2. DIFFERENTIAL CONDUCTANCE 97
as Vds increases. Quantitatively, two regions are divided around Vds ∼ 10 mV, which
corresponds to the energy spacing. In this model, one possible interpretation can
be as follows: When Vds is smaller than ∆E/e, the coherence length of the electron
wavepackets is longer than the tube length, L. In this region, the SWNT operates
as an isolated zero-dimensional quantum dot between two leads. For Vds > ∆E/e,
the coherence length of electron wavepackets is shorter than L. In this limit, each
wavepacket propagates through the one-dimensional conductor, and the oscillating
period increases. This phenomenon may indicate the correlated electrons since iso-
lated wavepackets are likely to experience unscreened Coulomb interactions. [91].
0.52
0.50
0.48
0.46
0.44
0.42
0.40
0.38
dI/
dV
(/
2GQ
)
403020100 Vds (mV)
Figure 5.12: The differential conductance from a 360 nm-long SWNT device at Vg = -5 V. Experimental data are blue circles and the theoretical fitting of the single-channeldouble-barrier structure model is in red.
Fabry-Perot Interferometer
Liang et. al. captured the wave nature of electrons through a isolated nanotube as an
electron waveguide. Two interfaces at metal and tubes are one-to-one correspondence
98 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
(b)
(a)
a1r c1l
d1rb1l
c1r a1l
b1rd1l
a2r c2l
d2rb2l
c2r a2l
b2rd2l
a1r c1l
d1rb1l
c1r a1l
b1rd1l
a2r c2l
d2rb2l
c2r a2l
b2rd2l
Figure 5.13: (a) Diagram of two-channel double barrier system. (b) Two-dimensionalimage plot of differential conductance versus drain-source voltage in y-axis and back-gate voltage in x-axis.
5.2. DIFFERENTIAL CONDUCTANCE 99
to partially reflecting mirrors in a FP interferometer [16]. Similar to photons in
the FP cavity, electrons would experience multiple reflections between two barriers
separating the metal reservoirs from the SWNT before escaping. The theoretical
model is based on the Landauer-Buttiker formalism with four conducting channels
including spin and orbital degeneracies. The authors in Ref [16] dealed with two
spin-degenereate transverse channels by lifting the orbital degeneracy although they
did not address what causes the orbital degeneray lifted. The approach is to establish
three scattering (S) matrices at left and right interfaces and inside the tube denoted
as SL, SR, and SN . The sketchy of the model is drawn in Fig. 5.13(a). The numbers
1 and 2 mean the spin-degenerate channels, yielding 4×4 scattering matrices. As
mentioned above, metallic infinite SWNTs have dominantly forward scatterings while
backscattering and interbranch scatterings are prohibited due to a big momentum
transfer 2kF between two K points and the symmetry between two orthogonal π and
π∗ orbitals. Therefore, the backscattering which leads to resistance increase occurs
possibly only at the interfaces between metal electrodes and the tube. Explicitly,
the incoming and outgoing operators for two channels in a system have the following
notation in Fig. 5.13(a). Two scattering matrices SL and SR contain this assumption
such that
d1r
d2r
b1l
b2l
= exp
i
0 0 r2 r1eiδ1
0 0 r1eiδ1 r2e
iδ2
r2 r1e−iδ1 0 0
r1e−iδ1 r2e
−iδ2 0 0
a1r
a2r
c1l
c2l
, (5.4)
and
b1r
b2r
d1l
d2l
= exp
i
0 0 r2 r1e−iδ1
0 0 r1e−iδ1 r2e
−iδ2
r2 r1eiδ1 0 0
r1eiδ1 r2e
iδ2 0 0
c1r
c2r
a1l
a2l
. (5.5)
All scattering matrices should satisfy the Unitarity, so they are taken in exponential
forms under Born approximation. r1 and r2 are intra-mode and inter-mode reflection
coefficients. δ1 and δ2 are phase shift for intra-mode and inter-mode scatterings.
100 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
These represent weak backscattering and inter-mode mixing. It is fair to assume
symmetric barriers so that the four parameters are commonly shared in both matrices.
In addition, the multiple reflection part in the middle comes in SN is
c1r
c2r
c1l
c2l
=
eiφ1 0 0 0
0 eiφ2 0 0
0 0 eiφ1 0
0 0 0 eiφ2
d1r
d2r
d1l
d2l
. (5.6)
In the section of the tube, since inter-mode mixings are not allowed. φ1 and φ2 are
accumulated phase of individual channels. The total scattering matrix ST is a matrix
product of all three matrices, ST = SL
⊗
SN
⊗
SR. The electronic FP interferometer
is written with the ST matrix,
b1r
b2r
b1l
b2l
=
t1r,1r t1r,2r r1r,1l r1r,2l
t2r,1r t2r,2r r2r,1l r2r,2l
r1l,1r r1l,2r t1l,1l t1l,2l
r2l,1r r2l,2r t2l,1l t2l,2l
a1r
a2r
a1l
a2l
. (5.7)
The differential conductance is
dI
dVds
=2e2
h
2∑
i=1
|ti|2 =2e2
h
2∑
i=1
Tr(S†TST ). (5.8)
Similar to the double-barrier case, the energy dependence is included in the phase
parameters. Figure 5.13(b) presents the two-dimensional image plot from theoretical
modelling with r1 = 0.5, r5 = 0.25, δ = 0.4, δ = 0.95 given in Ref. [16] and for L = 360
nm. Since the model is valid for weak backscattering, the values of r1 and r2 should
not be big, whereas two other fitting parameters are randomly chosen until the fit is
close to the data. It would be the drawback of the model in that there are too many
free parameters which are not experimentally accessible.
5.2. DIFFERENTIAL CONDUCTANCE 101
¡L2
L2
glead = 1 glead = 1g < 1
¡1 1
1 2
¡L2
¡L2
L2
L2
glead = 1glead = 1 glead = 1glead = 1g < 1g < 1
¡1¡1 11
1 2
Figure 5.14: Illustration of the TLL model on a SWNT device.
Tomonaga-Luttinger Liquid
The third attempt to understand the empirical behaviors in dI/dVds is motivated by
the fact that SWNTs have exhibited the features of strong correlations among charge
carriers in experiments [23] and in theories [65, 66, 91]. SWNTs are one of the ideal
systems to investigate one-dimensional physics since their geometric ratio between ra-
dius and length ranges from 100 to 106. Plus, a long-range Coulomb interaction makes
metallic SWNTs to be a Tomonaga-Luttinger liquid (TLL), a non-Fermi liquid. The
TLL theory is a framework to describe the low-energy properties of one-dimensional
conductors [62, 63, 92, 93]. The interaction strength in SWNTs is estimated theoreti-
cally with the a TLL interaction parameter g ∼ 0.2 − 0.3. In comparison with g = 1
for the non-interacting Fermi gas, SWNTs are strongly correlated systems, in which
Coulomb interactions play a crucial role in the current flow.
The ballistic SWNT device including metal electrodes are theoretically modelled
as an infinite one-dimensional conductors with inhomogeneous the TLL parameter
g(x). The interaction is assumed to be strong in the SWNT (0 < g < 1) and weak
in the higher dimensional metal reservoirs (g = 1) for metals [91, 94]. The schematic
of the model is illustrated in Fig. 5.14. The TLL without the barriers is described by
102 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
the bosonized Hamiltonian [65]
Hswnt = (vF/2π)∑
a
∫
dx[(∂xφa)2 + g−2
a (x)(∂xθa)2]
, where θa(x) and πa(x) = −∂xφa/π are conjugated bosonic variables, i.e. [θa(x), πb(x′)] =
iδabδ(x− x′). The four conducting transverse channels of the SWNTs in the FL the-
ory are transformed to four collective excitations in the TLL theory: one interacting
collective mode (a = 1, ga ≡ g) of the total charge and three neutral non-interacting
collective modes (a = 2−4, ga = 1) including spin. There are two distinct propagating
velocities vc = vF/g and vF . The inter-channel and intra-channel scattering occurs at
the barriers. These modes are partially reflected at the two barriers. The backscatter-
ing is supposed to be weak enough that the backscattering Hamiltonian term can be
treated as a perturbation. Non-equilibrium situation is treated within the Keldysh
formalism. The transport properties are computed from correlation and retarded
Green’s functions [94]. Three non-interacting modes encounter backscattering at the
physical barrier, whereas the interacting mode encounters the momentum-conserving
backscattering due to g mismatch at the interfaces as well as the physical barrier
backscatterings. The theoretical differential conductance I ≡ e(2/π)θ1 = 2GQVds−IBwhere IB is given to leading order in the backscattering amplitudes as [91,94]
IB =2e
πt2F
∑
b=1,2
Ub
∫ ∞
0
dteCb(t) sin
(
Rb(t)
2
)
sin
(
eVdst
~
)
, (5.9)
where tF = L/vF is the travelling time for a non-interacting mode along the SWNT
length L. The backscattered current IB consists of two contributions: the term pro-
portional to U1 represents the incoherent sum of backscattering events at the two
barriers and the term associated with U2 results in the FP oscillations due to the
coherent interference between backscattering events from different barriers. At high
Vds, the U1-term in Eq. (5.9) dominates and the oscillation amplitude decreases. U1
and U2 are independent of Vds, but U2 depends periodically on Vg. The interaction
parameter g is involved in the time integral through Cb(t) and Rb(t), which are cor-
relation and retarded functions, respectively. These Green’s functions contain a sum
5.2. DIFFERENTIAL CONDUCTANCE 103
over all four collective modes, and their forms are obtained at zero [91,94] and finite
temperatures [94]. Mathematically, the terms are proportional to the backscattering
amplitudes uαi of each mode (i = 1,2,3,4) at left and right barriers (α = l, r),
U1 ∝4∑
i=1
(uli)
2 + (uri )
2
U2 ∝4∑
i=1
2uliu
rj cos(
VgCL
e+ 2∆ij)
(5.10)
where C is the capacitance from the backgate and ∆ij the phases for inter- and
intra-channel mixing.
A simplified form of the differential conductance including Eq. (5.9) in the unit
of 4GQ is written with two Vds-dependent f1 and f2 functions,
dI
dVds
= 4e2
h[1 + U1 · f1(Vds, g) + U2 · f2(Vds, g)] . (5.11)
Both two backscattering contributions have the sine function of the drain-source volt-
age and contain the TLL parameter through Green’s functions.
In Fig. 5.15, the Vds dependent functions f1 and f2 are displayed for different g
values at T = 4 K and the tube length L is 360 nm. The function f1 exhibits the
power-law behavior for g less than 1 in Vds, and the function f2 exhibits oscillation
amplitudes due to the Fabry-Perot interference. The backscattering amplitudes are
renormalized by the interaction parameter g, reducing the oscillation amplitude in Vds.
Figure 5.16 (b) shows overall theoretical fitting plots combining two terms for g = 0.25
(red) and g = 1 (blue) with U1 = 0.14 and U2 = 0.1. The plot with g = 1 represents
the case that all four modes are non-interacting, thus the interference amplitudes
are constant regardless of the bias voltage size. Clearly, the theoretical differential
conductance trace for g = 0.25 has qualitative agreement with the experimental data:
the amplitude of the FP oscillation is damped at high Vds compared to that at low
Vds. The differences between the TLL theory and the experimental data are found
in that the overall conductance level in the data is much lower than the theory, and
the oscillation period in Vds increases in experiments. In addition, the asymmetry
104 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
403020100 Vds (mV)
(b)
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
403020100 Vds (mV)
g = 0.25 g = 0.75 g = 1
(a)
Figure 5.15: (a) The Vds-dependent f1(Vds, g, T ) at T = 4 K for g = 0.25 (red),g = 0.75 (green) and g = 1 (blue). (b) The Vds-dependent f2(Vds, g, T ) at T = 4 Kfor g = 0.25 (red), g = 0.75 (green) and g = 1 (blue).
in positive and negative Vds is vivid in experiments, but the cause is not definitely
understood yet.
To identify the TLL feature uniquely in experiments requires to increase Vds above
5.2. DIFFERENTIAL CONDUCTANCE 105
the levelspacing ~/2gtF. Note that the tendency of amplitude reduction in experimen-
tal data cannot be reproduced by the reservoir heating model [95] which asserts that
the dissipated power V 2ds(dI/dVds) leads to a bias-voltage dependent electron temper-
ature. We have tested this effect for the non-interacting case (g=1) in our theory
and have found that it causes a slight damping of the FP-oscillations (U2-term) but
the incoherent part (U1-term) is independent of temperature. The temperature effect
fails to account for the experimentally observed enhanced backscattering amplitude
at low Vds [94]. In addition, the conductance is relatively small (on the order of GQ)
so that heating effects should not be pronounced in the bias window considered.
5.2.2 Spin-Charge Separation
The TLL theory provided a clue to interpret the Vds-dependent differential conduc-
tance traces, a qualitative feature of interactions. Figure 5.17(a) explicitly shows
several traces in Vds at different values of Vg with the following pronounced features:
the oscillations in low Vds have different periods at different Vg and the period of
oscillations become elongated at high Vds.
First, consider the former feature: Vg-dependent oscillation periods. The theory
predicted the existence of two propagating velocities by means of the backgate volt-
ages. This feature is captured from the fact that the U2 backscattering term of FP
interference contains a cosine function of Vg. The oscillation patterns from the U2
term are governed by three non-interacting modes. Although the interacting charge
mode behaves differently from the other three modes, the modification of the inter-
acting charge mode is too slight to be recognized. On the other hand, the U1 term
shows the enhanced backscattering amplitude at low Vds with the power-law scalings
and contains the interference of only the charge mode due to the mismatch of g at the
interface (Fig. 5.15(a)). If the interaction-induced interference can be observed, the
quantitative information of the TLL behavior in ballistic SWNTs is obtained. The
key to observe the charge mode interference is to remove the FP contribution which
masks the interaction effect.
Since Vg determines the location of the Fermi level in the bandstructure, the choice
106 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
0.38
0.36
0.34
0.32
0.30
0.28
dI/d
V
-40 -20 0 20 40 V
ds(mV)
(a)
(b) 1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
dI/
dV
ds
-40 -20 0 20 40
Vds (mV)
Figure 5.16: Differential conductance versus Vds at a certain Vg. (a) Experiment and(b) Theory based on TLL for g = 0.25 (red) and g = 1 (blue).
5.2. DIFFERENTIAL CONDUCTANCE 107
0.38
0.36
0.34
0.32
0.30
dI/
dV
ds
-40 -20 0 20 40 V
ds (mV)
-9V
-8.3V
-7.7V
1.00
0.95
0.90
0.85
0.80
0.75
0.70
d
I/d
Vd
s
-20 -10 0 10 20 V
ds (mV)
(a)
(b)
(c)
Figure 5.17: Differential conductance versus Vds at different backgate voltages forexperiment results (a), the theoretical plots from non-interacting Fermi-liquid theory(b) and from the Tomonaga-Luttinger liquid theory (c).
108 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
of U2 sets up whether the maximum or minimum dI/dV starts at the zero Vds shown
in Fig. 5.17(c). Furthermore, note that the FP oscillations are absent at a certain Vg
according to the cosine function of Vg. It means that the backscattering contributions
are π/2 out of phase, leading to destructive interference. It is speculated that the
contribution from ac signals on top of Vds/2 and −Vds/2 from the Fermi level add
up constructively at the node of cosine function in Vg, whereas they cancel out at
the antinode of cosine function in Vg since the k values within the ac signal window
involve in transport processes. In the non-interacting two-channel model, the effect of
Vg shows a simple oscillating amplitude behavior in all regions of Vds in Fig. 5.17(b).
On the other hand, the outcome of the TLL shows the dramatic change such that
the weak oscillation emerges at U2 = 0. The position of the peak sits in higher Vds
compared to the non-zero U2. From the period of the interference patterns, the values
of the mode velocities are extracted due to the fact that the periods in Vds relates to
the travelling distance (l) and velocity (v) such that e∆Vds = hv/l. With vc = vF/g
and the tube length L, the TLL parameter g is written as
g =∆Vds(U2 6= 0)
2∆Vds(U2 = 0). (5.12)
Sorting the dI/dV traces into two groups and measuring the periods, an indication of
this effect is found by comparing the primary periods (2gtF if U2 ∼ 0 and tF when U2
is maximal) of these traces, which gives g ∼ 0.22. This fact hints that the total charge
mode has its own velocity vF/g in the SWNT. In the literature, spin-charge separation
in semiconducting wires has been recently observed by mapping the distinct charge
and spin mode velocities [96].
The latter signature of the longer period at high Vds is beyond our theory, but
is likely to be caused by a strong barrier asymmetry at high Vds which would also
suppress the U2-term. The ratio of primary and elongated periods along Vds relates
to g ∼ 0.22. To reach a conclusive claim, further experiments focusing on this aspect
should be performed.
5.3. LOW-FREQUENCY SHOT NOISE 109
5.3 Low-Frequency Shot Noise
5.3.1 Experiment Setup
Two-terminal shot noise measurements were implemented using all technical strate-
gies presented in Chapter 4. Figure 5.11 illustrates the simplified diagram of the
SWNT shoe noise measurement setup. The design of the experiments has required
to consider several technical issues: the shot noise measurement frequency and cali-
bration.
1/f noise crossover
To determine the shot noise measurement frequency f is one of the first things to
be considered before implementation. The upper bound of f is, in principle, limited
by the energy scaling comparison among eVds, kBT, and ~ω, where ω is the angular
frequency, equal to 2πf . For SWNTs, Vds can be applied up to 40 ∼ 50 mV without
saturating the current. Thus the biggest energy scale is eV ∼ 40 meV. The thermal
energy at 4 K is 0.33 meV. Therefore, any frequency as ~ω is lower than kBT works
well, which corresponds to f ∼ 400 GHz. 400 GHz is in the microwave regime. To ex-
ecute any measurements at such high frequency in the microwave regime is extremely
difficult because there are very delicate technical treatments involved. Therefore, the
practical upper bound is taken by the wavelength of chosen frequency. As long as
the wavelength is much larger than the system size (rigorously, the system size is
smaller than a fourth of the wavelength as a rule of thumb), subtle microwave issues
can be ignored. Since the cryostat including wirings is around 2 m, the wavelength 8
m corresponds to around 37.5 MHz. Roughly speaking, tens of MHz range is a good
upper limit. The lower bound of the measurement frequency is set by the 1/f noise
crossover mentioned earlier in Chapter 3.
The 1/f noise of carbon nanotubes has been measured in various forms: SWNT
mats and isolated SWNTs [97], two-crossing MWNTs [98], SWNTs at room tempera-
tures (RT) [99] and MWNTs at low temperatures [100]. The 1/f noise measurements
were executed with a current bias scheme at three temperatures: RT, 77 K(liquid
110C
HA
PT
ER
5.
SIN
GLE
-WA
LLE
DC
AR
BO
NN
AN
OT
UB
ES
Gv
Resonant
Circuit
RPD>>RSWNT
Signal
DCVg
RSWNT
-20V
LED
Vdc
Vac
SWNTVdc
Vac
+
+Cparasitic
*
#
( ) 2 Lock-InGv
Resonant
Circuit
Resonant
Circuit
Resonant
Circuit
RPD>>RSWNT
Signal
DCVg
RSWNT
-20V
LED
Vdc
Vac
Vdc
Vac
SWNTVdc
Vac
Vdc
Vac
+
+Cparasitic
*
#
( ) 2 Lock-In( ) 2 Lock-In
Figu
re5.18:
Sch
ematics
ofSW
NT
shot
noise
measu
remen
tsetu
p.
5.3. LOW-FREQUENCY SHOT NOISE 111
10-16
10-15
10-14
10-13
10-12
10-11
SV (
V2 /
Hz
)
103
104
105
106
Frequency (Hz)
I = 1125 nA @ T = 77 K
I = 1209 nA @ T = 293 K
Figure 5.19: The 1/f noise crossover of SWNT devices for two temperatures 293 Kand 77 K.
nitrogen) and 4 K. The representative 1/f voltage noise spectral densities are shown
in Fig. 5.19. The calibration of 1/f noise measurements can be achieved by comparing
with the theoretical thermal noise because without the bias, the only possible noise
source is the thermal noise. The red straight lines are thermal noise voltage 4kBTR
at T = 283 K and T = 77 K. with measuring linear resistance R at given tempera-
tures. Indeed, both noise traces without any bias at RT and 77 K sit exactly on top
of theoretical values. As the non-zero current is applied to the SWNT, the excess 1/f
noise signals emerge. The total noise spectral density is
SV,Tot = AV 2
fα+ 2eIR2 + 4kBTR. (5.13)
112 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
The corner frequency fc is found as the pink noise is equal to the white noise level,
fc =[
AV 2(2eIR2 + 4kBTR)−1]1/α
. (5.14)
Fitting the 1/f noise trace with the Hooge’s formula gives constant A and the power
exponent α of frequency. Experimentally, the conservative value of fc is estimated
from setting the 1/f noise equal to the thermal noise since the white noise level is
higher with adding the shot noise. The values of fc is around 5 MHz at RT and
becomes lower to 1 MHz at 77 K. The origin of the 1/f noise would relate to the
trapping impurities in oxides. They may be frozen at lower temperature, reducing
the 1/f noise and fc. From the lower bound of the fc and the upper bound of 40
MHz, the range of the shot noise frequency is tens of MHz. The actual measurement
implementation of the frequency choice is done by a tank circuit with passive com-
ponents, inductors and capacitors. The resonant frequency at RT in the experiment
is aimed at 20 MHz and it is reduced down to 14 ∼ 15 MHz at 4 K due to the
parasitic capacitance. The RT bandpass filter is placed with the low and high cutoff
frequencies of 12 and 21.4 MHz to carry the noise signals to the outside world.
A Full Shot Noise Source
The shot noise measurements were performed by placing two current noise sources
in parallel: a SWNT device and a full shot noise generator. The role of the full
shot noise is to calibrate the circuit. Using a cryogenic amplifier boosts the signal-
to-noise ratio but adds the complexity of the calibration due to transfer function of
the amplifier. The ideal full shot noise source has the particle statistics governed by
the Poisson distribution. A pair of a light emitting diode (LED) and a photodiode
(PD) is selected as a full shot noise candidate. PD converts the photon energy by
accepting photons emitted from the LED into electrical current. In particular, as
long as the lower coupling efficiency between the LED current and the PD current is,
the shot noise of the PD current is closer to the ideal full shot noise. The coupling
efficiency between two currents is measured at RT and 4 K. Photons from the LED is
proportional to the LED current. At 4K, the ratio of LED and PD current is about 0.1
5.3. LOW-FREQUENCY SHOT NOISE 113
1400
1200
1000
800
600
400
200
0
IP
D (
nA
)
400x103
3002001000 ILED (nA)
T = 4.2 Kα ~ 0.38 %
2000
1500
1000
500
0
IP
D (
nA
)
1.6x106
1.41.21.00.80.60.40.20.0 ILED (nA)
T = 293 Kα ~ 0.13 %
(a)
(b)1400
1200
1000
800
600
400
200
0
IP
D (
nA
)
400x103
3002001000 ILED (nA)
T = 4.2 Kα ~ 0.38 %
2000
1500
1000
500
0
IP
D (
nA
)
1.6x106
1.41.21.00.80.60.40.20.0 ILED (nA)
T = 293 Kα ~ 0.13 %
(a)
(b)
Figure 5.20: The coupling efficiency α between ILED and IPD at (a) T = 293 K and(b) T = 4 K.
114 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
(a)
(b)
250
200
150
100
50
0
SP
D (a
.u.)
2000150010005000 IPD (nA)
T = 293 KRinput = 10 kΩ
400
300
200
100
0
SP
D (a
.u.)
160012008004000 IPD (nA)
T = 4.2 KRinput = 10 kΩ
(a)
(b)
250
200
150
100
50
0
SP
D (a
.u.)
2000150010005000 IPD (nA)
T = 293 KRinput = 10 kΩ
400
300
200
100
0
SP
D (a
.u.)
160012008004000 IPD (nA)
T = 4.2 KRinput = 10 kΩ
Figure 5.21: Full Shot Noise from the LED/PD pair (a) T = 293 K and (b) T = 4 K
5.3. LOW-FREQUENCY SHOT NOISE 115
- 0.4 %, eliminating completely the shot noise squeezing effect due to constant current
operation [101]. The full shot noise is confirmed with a constant input resistor R =
10 kΩ. The shot noise signal after the tank circuit is fed into the cryogenic amplifier
followed by the bandpass filter, a square-law detector and a lock-in amplifier. The
signals of the lock-in amplifier at both RT and 4 K exhibit the linear relation of the
PD current as expected (Fig. 5.21). The slope of the noise and the current is related
to the following quantities: the elementary charge e, resistance, the equivalent noise
bandwidth and the gain of the amplifier. The output spectrum before the square
law detector is captured with a HP8561E spectrum analyzer, from which the voltage
gain and the area of the spectrum are measured. Then, the unknown value from the
slope is the size of the elementary charge. With a R = 10 kΩ at RT and 4 K shown
in Fig. 5.21, the linear regression analysis tells the slopes of the full shot noise are
0.2544 ± 0.000543 (0.11853±0.0003) at 4 K (RT). Together with the voltage gain Av
= 1111.6037 (676.005) and the equivalent noise bandwidth BW = 1.330 MHz (1.584
MHz) are obtained from the spectrum, the elementary charge is 1.52 × 10 −19 (1.61
× 10 −19) with a 5 % (0.6 %) error to the ideal value e = 1.6× 10 −19.
5.3.2 Shot Noise and Fano factor versus the Drain-Source
Voltage
Two-terminal shot noise measurements are performed at a particular Vg value. Fig-
ure 5.22 shows a typical shot noise SSWNT versus Vds on a log-log scale for a particular
Vg. SSWNT (dot) is clearly suppressed to value below the full shot noise SPD (triangle).
It means the charge flow is regulated further beyond Poisson statistics and it suggests
that the relevant backscattering for shot noise is indeed weak. It can be understood
as the partition noise due to the partial transmission between the metal reservoir and
the tube. Another noticeable feature between two noise graphs is clearly different
scaling slopes versus Vds.
The Green’s function theory of the TLL model is extended to calculate the shot
116 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
2.5
2.0
1.5
1.0
0.5
lo
g (
SP
D, S
SW
NT )
1.61.41.21.00.80.60.40.20.0 log ( V
ds )
Figure 5.22: A representative log-log plot of low-frequency shot noise from theLED/PD pair (upside-down triangle) and the SWNT (diamond) as a function ofVds. The straight line is the outcome of linear regression analysis.
noise spectral density in the zero-frequency limit,
Sswnt(ω) =
∫
dteiωt〈δI(t), δI(0)〉
with δI(t) = I(t)− I the current fluctuation operator and · · · the anticommutator
[94]. In this model, the SWNT noise is expressed as Sswnt = 2e coth(eVds/2kBT )IB +
4kBT (dI/dVds − dIB/dVds), becoming SSWNT = 2eIB for eVds > kBT . This simple
result indicates that the charge carriers in low frequency transport processes are
electrons not fractional charges in the TLL. It is argued rather intuitively that the
carriers in the metal electrodes are recovered to electrons for a long average over the
transit time. IB is computed by integrating backscattering contribution in dI/dVds
with respect to Vds.
5.3. LOW-FREQUENCY SHOT NOISE 117
IB ≡∫ ∞
0
dVds
(
4e2
h− dI
dV
)
=
∫ ∞
0
dVds(U1f1(Vds, g, T ) + U2f2(Vds, g, T )).
(5.15)
Figure 5.23 displays the g-value effect on two backscattering contributions. Again
the U1 term shows the asymptotic behavior in Vds. Without interaction (g = 1),
the backscattered current is in a perfect linear relation with Vds. As soon as the
interaction turns on, the correction due to interactions deviates from the linearity.
The deviation becomes larger for the smaller g. The interference in the U2 term
remains in IB. The values of the fitting parameters U1 and U2 at 4 K for a 360
nm SWNT device produces theoretical shot noise in red together with experimental
data in blue. The TLL theory explains the interference pattern in the noise very well
(Fig. 5.24).
Moreover, the asymptotic behavior of IB from the dominant U1-term when eVds >
~/2gtF yields the power-law relation IB ∼ V 1+αds with α = −(1/2)(1−g)/(1+g). Note
that the power-law scaling exponent α is uniquely determined by the TLL parameter
g. The power exponent in this sample (Fig. 5.22) at Vg = −7.9 V is - 0.36 for the
SWNT, whereas α for the PD is zero. The average values of power-law exponent over
seven different gate voltages are estimated to be α ∼ - 0.31± 0.027, corresponding
to g ∼ 0.25 ± 0.049, a value which closely matches the experimental value g ∼ 0.22
obtained by the differential conductance oscillation mentioned above and also the
theoretical g-value for SWNTs.
The Fano factor F (I) was obtained at each current value by taking the ratio of
the SWNT noise and the PD noise, and it is presented on a log-log scale in Fig. 5.25.
The theoretical Fano factor F (I) ≡ §SWNT/2eI also manifests an asymptotic power
behavior in the high bias regime (eVds > ~/2gtF ), F ∼ V αds, assuming the backscat-
tered current is smaller than its ideal value 2GQVds. A linear regression analysis of the
Fano factor F with Vds, therefore, is another means to obtain the g value. The Fano
118 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
40
30
20
10
0403020100
Vds
(mV)
g = 0.25 g = 0.75 g = 1
(a)
(b)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
403020100 V
ds (mV)
Figure 5.23: (a) The integration in Vds of the Vds-dependent f1(Vds, g, T ) at T = 4 Kfor g = 0.25 (red), g = 0.75 (green) and g = 1 (blue). (b) The integration in Vds ofthe Vds-dependent f2(Vds, g, T ) at T = 4 K for g = 0.25 (red), g = 0.75 (green) andg = 1 (blue).
5.3. LOW-FREQUENCY SHOT NOISE 119
25
20
15
10
5
SS
WN
T (
a.u
)
403020100V
ds (mV)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
§ EXP
Theory
14.01 =U
1.02 =U
KT 4=
nm360=L
Fitting Parameters
25
20
15
10
5
SS
WN
T (
a.u
)
403020100V
ds (mV)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
§ EXP
Theory
14.01 =U
1.02 =U
KT 4=
nm360=L
14.01 =U
1.02 =U
KT 4=
nm360=L
Fitting Parameters
Figure 5.24: The experiment data (blue square) of the SWNT noise with the theo-retical fitting plot(red).
factor F for g = 0.25 (red) and g = 1 (yellow) is displayed n Fig. 5.25 (a). The log-
log scale presentation of the Fano factor is easy to appreciate the power-law scalings
and the linear scale presentation of F shows well that theoretical model for g = 0.25
gives the oscillatory behavior which matches well with experimental data (diamonds).
The stiffer slope (α) corresponds to stronger electron-electron interaction. The mean
value of the exponent α and the inferred g derived over seven Vg values are α = - 0.33
± 0.029 and g = 0.22 ± 0.046 respectively for this particular sample. We find that
the measured exponents α and inferred g values from the spectral density Sswnt and
the Fano factor F from four different devices with various metal electrodes (Ti/Au,
Ti-only, Pd) show similar statistics α ∼ −0.31±0.047 and g ∼ 0.26±0.071 as derived
from several Vg values for each sample. We stress that the non-linear decay of the
experimental F along Vds indeed starts at a voltage scale log(~/2getF) ∼ 0.61 for
g ∼ 0.18 in Fig. 4 as a manifestation of a collective electron effect.
The inferred g values from the exponents of the spectral density SSWNT and the
120 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
(a)
(b)
0.30
0.25
0.20
0.15
0.10
Fan
o f
ac
tor
403020100Vds (mV)
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4 lo
g (
Fan
o f
acto
r)
1.51.00.50.0
log ( Vds )
Figure 5.25: A representative log-log plot of Fano factor (open diamond) ob-tained from experiments against Vds together with theoretical theoretical fitting ofTomonaga-Luttinger liquid theory for g = 1 (straight line) and g = 0.25 (dottedline). The broken line on the experimental data represents the power-law scalinganalysis.
5.3. LOW-FREQUENCY SHOT NOISE 121
Sample Electrodes g(Noise) g(Fano factor)1 Ti/Au 0.25 ± 0.13 0.19 ± 0.072 Ti/Au 0.31 ± 0.086 0.31 ± 0.203 Pd 0.36 ± 0.18 0.11 ± 0.0784 Pd 0.28 ± 0.12 0.23 ± 0.057
Table 5.1: The g values from the power-law scaling analysis of four samples.
Fano factor F from four different devices with various metal electrodes (Ti/Au, Ti-
only, Pd) are listed in the Table 1. The overall statistical result is summarized as
α ∼ - 0.31 ± 0.047 and g ∼ 0.26 ± 0.071 as derived from several Vg values for each
sample. The experiments from the shot noise and the Fano factor are consistent with
each other as well as theoretical prediction.
5.3.3 Fano Factor versus Transmission Probability
Ballistic Phase-Coherent Picture
Previous subsection has focused on the trend of the Fano factor as a function of
Vds. In this section, the absolute values of the Fano factor are considered in terms
of the transmission probabilities. The definition of the transmission probability T
is not very obvious from experimental data since two-terminal resistance in SWNTs
contains not only the intrinsic resistance but also the physical contact resistance
and the lead resistance. Since the contributions of different resistance sources are
not distinguished, experimentally the possible way to estimate T is the ratio of the
measured resistance and the quantum unit of resistance.
Model I : One Transmission Probability T Mesoscopic partition noise with a
lumped elastic scatterer with a transmission probability T is a form of SI = 2eI(1−T ).
In this case, the Fano factor is simply reduced to (1-T) because F = SI/2eI = 1−T .
Model II : Two-Channel Model Suppose two differential transmission probabilities
T1 and T2 exist in the system. It can be either two channels in the system or one
channel in the double barrier system. In this case, the total transmission probability
T is related to T1 and T2 such that T = (T1 + T2)/2. The shot noise SI with two
122 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
values is written as SI = eGQV (T1(1 − T1) + T2(1 − T2)) where the current I is
I = GQV T = GQV (T1 + T2)/2. This can be expanded into a n-channel model:
T =
∑ni=1 Ti∑n
i=1
, (5.16)
F =
∑ni=1 Ti(1 − Ti)∑n
i=1 Ti
. (5.17)
In this model for a fixed T , F is not single-valued but it has the range bounded
by the maximum and minimum values from differentiating F with respect to T and
setting to zero. Table 5.2 summarizes the range of the allowed Fano factor values as
a function of T in the two-channel model:
F expressionsFmax = 1 − T for 0 ≦ T ≦ 1
Fmin = 1 − 2T for 0 ≦ T ≦ 1/2
= (2T−1)(1−T )T
for 1/2 < T < 1
Table 5.2: The relation between extreme values of F and the overall transmissionprobability T
Figure 5.26 (a) shows the experimental data as well as the above Fano factor
relations for one and two-channel cases. Experimental data are the Fano factor at
40 mV from five SWNT-devices denoted as different symbols. Given a device, many
points were taken by sweeping gate voltage. Some devices show the resistance change
over the gate voltage change. Apparently, experimental data are completely off from
any models in the ballistic phase-coherent conductors.
Phase-Incoherent Picture
Figure 5.26 clearly states that the Fano factor of experimental data is further sup-
pressed below the value of F with a lumped elastic scatterer in the ballistic regime.
Further suppression can be caused by correlations. In the literature, the effects on
5.3. LOW-FREQUENCY SHOT NOISE 123
1.0
0.8
0.6
0.4
0.2
0.0
F
ano
Facto
r
1.00.80.60.40.20.0
T (=G/(2GQ) )
1-T two-mode model
1.0
0.8
0.6
0.4
0.2
0.0
F
an
o F
acto
r
1.00.80.60.40.20.0
T (=G/(2GQ) )
distributed elastic scattering
distributed inelastic scattering
incoherent double-barrier model incoherent many-barriers model
(a)
(b)
Figure 5.26: Fano factor versus transmission probability taken at Vds = 40 mV fromfive SWNT-devices (filled symbols) at varying Vg values. (a) Ballistic phase-coherenttransport theory for one-(dark blue straight) and two-channel (dotted area) models.(b) Phase-incoherent picture theory for distributed elastic (square) and inelastic (dia-mond), incoherent double-barrier (light green straight) and many-barrier model (darkgreen straight)
124 CHAPTER 5. SINGLE-WALLED CARBON NANOTUBES
the suppressed shot noise of disorder has been studied both in experiments and in
theory [95, 102–110]. Considering the involvement of distributed elastic and/or dis-
tributed inelastic scatterers in transport processes, such systems are no longer in the
ballistic regime but rather in the diffusive regime. Theorists have showed that a dis-
ordered phase-coherent conductor exhibits one-third of the Poisson shot noise value
using the random-matrix theory [103]. The degree of suppression in the Fano factor
value (1/3) is also predicted in a semi-classical picture without phase coherence but
allowing to have the fluctuations of the distribution function [106]. Liu studied several
models based on the previous pictures by quantum Monte Carlo simulation in order
to produce the relation of the Fano factor and the transmission probabilities plot-
ted in Fig. 5.26 (b) [111, 112]. He tracked down the electron phase according to the
sources of scattering: elastic scatterers reverse the momentum of electrons whereas
inelastic scatterers lose the energy of electrons. Similar to the literature, he found
that distributed elastic scatterers reduced the shot noise by 1/3 from the Poisson
value as the transmission probability approaches to 0. However, distributed inelastic
scatters further suppress the shot noise close to 0 due to the complete heat removal
by phonons such as macroscopic resistor.
n-barrier models in the phase-incoherent picture were simulated [112]. For n
identical barriers with the same transmission probability t, the total transmission
probability T is expressed as T = t/(t + n(1 − t)). For this T , the Fano factor F is
estimated as
F =1
3
(
1 +n(1 − T )2(2 + (3n− 2)T ) − (nT )3
n3
)
.
In the case of a double barrier n = 2, F is reduced to F = (1/2)(1 − T )(1 + T ).
In Fig. 5.26(b), the light green line says F approaches to 0.5 as T decreases, while
the case of many-barrier structure yields F to be 1/3.
Unfortunately, all above models do not match any point of experimental data in
terms of the degree and the trend of the suppressed Fano factor along the transmission
probability. Therefore, the strongly suppressed Fano factor in SWNTs is due to some
other correlations among charge carriers. To understand the absolute value of the
5.4. SUMMARY 125
Fano factor in non-chiral Luttinger liquid conductors such as SWNTs is missing at
present and this is an interesting problem to study further.
5.4 Summary
Chapter 5 has discussed the low temperature electrical properties of metallic SWNTs.
The discussion started from the bandstructure of SWNTs in terms of the tight-binding
approximation to understand electronic properties. Focusing on metallic SWNTs,
the conductance and the current fluctuations have been measured. The experimental
outcome has exhibited unique features originated from correlations among charge car-
riers. The experimental data have been examined by both non-interacting and inter-
acting theories. A non-interacting picture may describe conductance data in the low
bias regime; however, it fails to explain high-energy data. The Tomonaga-Luttinger
liquid theory has explained the qualitative trend of conductance as a function of the
drain-source voltage. In addition, it has captured the quantitative information of the
strong electron-electron interactions both in conductance and the shot noise quanti-
ties. The strength of the interactions is parameterized by g, which has been obtained
from the conductance period at various Vg and power-law scaling exponents from
both shot noise and the Fano factor. The g values are consistent from two indepen-
dent measurements. This work contributes the first quantitative investigation of TLL
interaction effects in the shot noise of well-contacted SWNTs.
Chapter 6
Quantum Point Contact
The modern world,
despite a surfeit of obfuscation, complication, and downright deceit,
is not impenetrable, is not unknowable,
and – if the right questions are asked –
is even more intriguing than we think.
All it takes is a new way of looking.
− Steven D. Levitt
6.1 Two-Dimensional Electron Gas
A two-dimensional electron gas (2DEG) has emerged along the route to improve the
functional capacities of semiconductor transistor. It is a material where electrons’
free motion is only possible in a two-dimensional plane. It was found early on inside
silicon-based MOSFETs (metal-oxide field effect transistor), where electrons beneath
the gate oxide were trapped at the semiconductor-oxide interface. A 2DEG is also
formed in other elemental and compounds semiconductors, germanium, gallium ar-
senide (GaAs) and aluminum arsenide (AlAs). For compound semiconductors, the
2DEG is embedded in a heterojunction where two different materials face. The good
126
6.1. TWO-DIMENSIONAL ELECTRON GAS 127
example is a doped GaAs/AlGaAs heterostructure and the 2DEG is placed in the
GaAs layer according to the bandgap arrangement of GaAs and AlGaAs.
This new structure has endowed physicists with many opportunities to explore
the low dimensional physics. Further spatial confinements allow us to create one-
dimensional and zero-dimensional structures with advanced microfabrication tech-
nologies. The confinement is directly responsible for quantum-size effect. The 2DEG
has marked a new era in condensed matter physics since it reveals novel quantum
phenomena due to low dimensionality (example: quantum Hall effect) and also serves
a test bed of quantum mechanics postulates. The field has been very productive to
understand the following basic features: universal conductance fluctuations [?, 113],
localization [114], quantized conductance [45,115], Aharonov-Bohm interference [116],
Coulomb Blockade [117,118], Kondo effect [119,120], integer quantum Hall effect [121],
fractional quantum Hall effect [122] and more. In addition, a dream to establish solid-
state qubits for quantum computation in a 2DEG is actively pursued at present [49].
In addition, the quality of 2DEGs determines the transport regimes among dissipative,
diffusive and ballistic ones. This section discusses the principle of a 2DEG formation
in a GaAs/AlGaAs heterostructure, the realization of a GaAs/AlGaAs 2DEG and
the scattering mechanisms to explain the high quality of 2DEG traits.
6.1.1 Energy Band Profile
The understanding of how to form a 2DEG system in doped heterostructures starts
from energy band profiles. Let’s first consider undoped heterostructures composed of
GaAs and AlGaAs as common choices since their lattice constants are very close to
each other. Due to the almost perfect lattice match, the GaAs/AlGaAs heterostruc-
ture yields almost no strain at the interface. Often the strain becomes a possible scat-
tering source. The lattice constants of GaAs and AlAs are 0.56533 nm and 0.56611
nm respectively. In actual growth, instead of using AlAs, the alloy of GaAs and AlAs,
AlxGa1−xAs, is used for a larger bandgap layer. Lattice constant and bandgap energy
Eg are depending on the concentration (x) of Al in the alloy, and typically x is about
128C
HA
PT
ER
6.
QU
AN
TU
MP
OIN
TC
ON
TA
CT
Vacuum levels
i - GaAs
Eg = 1.414 eV
i – Al0.3Ga0.7As
Eg = 1.798 eV
Gro
wth
direct
ion
EcEv
E
(a)
i - GaAsEg = 1.414 eV
n – Alc.3Ga0.7As
Eg = 1.798 eV
2DEG
(b)Ev Ec
Vacuum levels
i - GaAs
Eg = 1.414 eV
i – Al0.3Ga0.7As
Eg = 1.798 eV
Gro
wth
direct
ion
EcEv
E
(a)
i - GaAsEg = 1.414 eV
n – Alc.3Ga0.7As
Eg = 1.798 eV
2DEG
(b)Ev Ec
Figu
re6.1:
(a)B
and
profi
leof
aG
aAs/A
lGaA
sheterostru
cture.
(b)
Ban
dprofi
leof
a2D
EG
embed
ded
ina
dop
edG
aAs/A
lGaA
sheterostru
cture.
6.1. TWO-DIMENSIONAL ELECTRON GAS 129
0.30. The lattice constant and bandgap energy for x < 0.45 have the following rela-
tions; lattice constant = 0.56533+0.00078x and Eg = 1.424+1.247x at the high sym-
metric Γ point from the band structure at room temperature. The bandgap energy
difference between two layers of GaAs and Al0.3Ga0.7As is ∆Eg = Eg(Al0.3Ga0.7As)
-Eg(GaAs) = 1.798 − 1.424 = 0.374 eV with lattice mismatch ∼ 0.000234nm by 0.04
% difference, which is ever better than the mismatch between GaAs and AlAs. The
question remains how to construct energy bands of two materials at the interface
where two layers meet. Anderson’s rule is a simple theory to answer the energy band
alignment question based on electron affinity from the vacuum levels [?, 61, 123]. It
says that the electron affinity of two materials should be in the same line. Following
Anderson’s rule, the GaAs/AlGaAs heterostructure energy band profile is shown in
Fig. 6.1 (a) by taking the affinity values of GaAs and Al0.3Ga0.7As, 4.07 eV and
3.74 eV. The affinity line match fixed the conduction band difference by ∆Ec ∼ 0.33
eV. Then, the remaining portion 0.04 eV from the bandgap energy 0.37 eV should
drop across the valence band, ∆Ev ∼ 0.04 eV. This heterojunction is a basic compo-
nent to grow quantum wells by sandwiching a narrow band GaAs with broad band
AlxGa1−xAs materials for exploring optical and electrical properties.
In the above heterostructure or quantum wells, all valence bands are occupied
at zero temperature while conduction bands are empty, which means systems are
insulating. In order to probe transport characteristics, charge carriers should be
introduced in the conduction band. There are several ways to do it: one is to use
optical pumping which excites electrons from valence bands to conduction bands
by light sources like lasers; another is to place electrically electrons or holes in the
bands, namely doping. A straightforward way to dope materials is to insert donors
or acceptors in designated locations; however, it is not really gainful since ionized
donors or acceptors become scatterers which ruin the quality , thus this way should
be avoided for the high mobility sample. The solution to this issue is modulation
doping. Its advantage is to separate ionized atoms from the interface, the location of
a 2DEG. In this case, the concentration of scatters is significantly reduced, yielding
the improved mobility of electrons in a 2DEG. The presence of ionized donors or
acceptors and migrating electrons or holes break neutrality in the growth direction.
130 CHAPTER 6. QUANTUM POINT CONTACT
This creates a built-in potential near the interface. As a result, the Fermi energy sits
in the conduction band, leaving the non-zero population of electrons in the potential
whose shape is more like a triangular well in a simple picture. Figure 6.1 (b) presents
a 2DEG within the GaAs conduction band, accordingly.
A Confinement Potential
A potential to trap a 2DEG near the interface is simply modelled as a triangular
well potential created by a linear electric field along the growth direction in z- axis,
Vz = eEz where e is the electron charge and E is the electric field magnitude. Note
that without losing a generality and with following the conventional choice, the growth
direction is designated in z-axis. The triangular well potential profile is valid only
for small z since the potential in the real heterostructure becomes flat far away from
the interface. Suppose z = 0 denotes the interface between GaAs and AlGaAs. The
corresponding Schrodinger equation with wavefunction in z-axis φ(z) is written as
[
− ~2
2m∗d2
dz2+ eEz
]
φ(z) = ǫφ(z). (6.1)
A strategy to solve Eq. (6.1) is to reduce it with dimensionless parameters which
replace z and ǫ. As a mathematical point of view, how to define such parameters
relies on pure dimensional analysis. Consider the following rearranged equation,
−d2φ(z)
dz2+
2m
~2eEzφ(z) =
2m
~2ǫφ(z). (6.2)
Since the wavefunction φ(z) has dimension in [length−1/2] but it is present in all
terms, its dimension can be neglected. The first term has the unit of [length−2] by
second derivative in z whereas the second is that of [length] from z. In order to
match dimension for both terms, the second term should be multiplied by something
of [length−3]. Thus, we can define the length scale z0 from the coefficients in front of
z to make a dimensionless parameter, z = z/z0, yielding z0 = (~2/2meE)1/3. A next
task is to express Eq. (6.2) in terms of z. Because of z = z/z0, the relations of the first
and second derivative between two parameters z and z are d/dz = (d/dz)(dz/dz) =
6.1. TWO-DIMENSIONAL ELECTRON GAS 131
(1/z0)(d/dz) and d2/dz2 = (1/z20)(d
2/dz2). Therefore, Eq. (6.2) is restated as
−d2φ(z)
dz2+ zφ(z) =
2mǫ
~2z20φ(z). (6.3)
Now, we are able to define the energy scale by ǫ = ǫ/ǫ0 where ǫ0 = ((eE~)2/2m)1/3.
Finally, the Schrodinger equation simplifies into
d2φ(z)
dz2+ (ǫ− z)φ(z) = 0. (6.4)
Note that indeed z0 and ǫ0 have the unit of [length] and [energy] automatically since
expression of z0 is in MKS[
kg·m2/s2·sJskgJ/m
]1/3
= [m3]1/3 = [m] and ǫ0 = eEz0 = [J ]. The
choice of parameters is just good. Equation (6.4) resembles Airy or Stokes equation,
d2y/dx2 = xy, which has two independent solutions called Airy functions Ai(x) and
Bi(x). The solutions are not simply expressed in an analytical form, but in modified
Bessel functions of order 1/3, whose numerical tables are available. Although both Ai
and Bi oscillate for a negative x, the asymptotic behaviors of Ai and Bi are distinct
in that Ai converges but Bi diverges as x grows in positive end. Thus, the solution to
Eq. (6.4) should be Ai not Bi due to convergence. Satisfying the boundary condition
φ(z = 0) = 0 at z = 0 yields quantized bound state energies ǫn = cn((eE~)2/2m)1/3.
The ground state energy has c1 ∼ 0.23381 from the exact Airy function evaluation.
Although this simple triangular-well model provides a qualitatively correct picture.
More realistic potentials would shape close to flat for large z including many-electron
effects. Variational Hartree approximations with Fang-Howard wavefunctions or nu-
merical Hartree self-consistent calculations have been pursued in these efforts [61].
6.1.2 Scattering Mechanism
Since the advent of 2DEGs in semiconductors, one of the main focus in material
preparation is to produce good quality 2DEGs. A good quality 2DEG is crucial
to study intrinsic features according to dynamics of electrons and their governing
principles. A figure of merit to identify the quality of a 2DEG is the mobility µ,
which provides information as to scattering processes in 2DEGs. The mobility µ is
132 CHAPTER 6. QUANTUM POINT CONTACT
a transport quantity and is given by the ratio of the drift velocity and an applied
electric field. Several lengthscales in mesoscopic systems introduced in Chapter 1 are
closely linked to this quantity µ. First of all, mean free path is lmfp = vF τ , where the
relaxation time τ is proportional to µ through µ = eτ/m∗. Phase-coherence length lφ
is also a function of µ in a slightly more complicated way via the relation lφ =√
Dτφ.
The mobility µ is related to the diffusion constant D via D = v2F τ/2 = v2
Fm∗µ/2e. In
this case, there is another time scale τφ, which represents the phase relaxation time.
Typically this τφ would be longer than τ . These two examples of µ involvement in
dynamics of electrons clearly state that µ is determined by scattering sources in the
2DEG.
There exist several sources to scatter electrons in semiconductor heterostructures:
defects of crystal structures, impurities in structure layers, and lattice vibrations
(phonons). At low temperatures, the last option, phonons would be negligible; how-
ever, other scattering centers will remain to limit the quality of systems. The previous
subsection briefly mentioned that the way to introduce doping contents greatly af-
fects scattering mechanisms. The standard reported number of the high mobility in a
2DEG is around µ = 107 cm2/Vs with a carrier density n2D = 2×1011 cm−2 achieved in
a delta-doped GaAs/AlGaAs heterostructure which has an inserting undoped spacer.
The spacer spatially separates ionized donors from a 2DEG. The values are truly
related to the structure design of a doping layer as well as the purity of molecular
beam epitaxy machine and target sources, which unintentionally introduce impurities
during growth processes.
Several research groups scrutinized scattering mechanisms as a function of electron
sheet density in a 2DEG [124–126]. The existent scattering sources in a delta-doped
heterostructure at low temperatures are divided into three categories: (1) remote
ionized donors, (2) unintentional background impurities in GaAs and the undoped
AlGaAs spacers and (3) interface roughness. The dominant scattering mechanisms
are different depending on electron density. At low n2D less than 5×1010 cm−2, the
major disorder comes from remote ionized impurites [124,125], whereas as n2D is more
than 5×1010 cm−2, homogenous background doping impurities are dominant with an
empirical trend µ ∝ nδ2D where δ ≈ 0.6 ∼ 1.1. The density-dependent trends in µ
6.1. TWO-DIMENSIONAL ELECTRON GAS 133
would be understood in a handwaving way that in the case of high n2D, more elec-
trons involve to screen effectively the potential induced by remote ionized impurities,
while background doping impurities remain unaffected by screening. According to
the literature, the 2DEGs with n2D ∼ 1012 cm−2 would have µ ∼ 106−7 cm2/Vs, so
that the corresponding mean free path is extended into 10 - 100 µm at submilli K
and clear features such as quantized conductance plateaus and quantum Hall effects
in a ballistic transport regime are exhibited.
6.1.3 Backgated 2DEG
From the previous discussion regarding scattering mechanisms, delta-doped 2DEG
structures at low temperature are superb to produce clean experimental results. Here
the devices we study have different methods to introduce conduction electrons in an
heterostructure not by doping but by backgating to n-type GaAs layer at the bot-
tom [127]. Such devices whose growth structure is sketched in Fig. 6.2 (a) are
designed for probing electron-electron interactions as a function of electron density
in the 2DEG because the strength of Coulomb interaction is directly influenced by
the number of electrons in a system. Figure 6.2 (b) shows a band diagram of induc-
ing carriers in a 2DEG by positive backgate voltages. The operation of a backgated
2DEG starts off the assumption that a surface charge density and its correspond-
ing electric field at the 2DEG surface are constant. As voltages get larger than the
threshold value Vbg,th applied to the n-type GaAs with respect to ground, the con-
duction band shifts downward, making the levels in a 2DEG align with the chemical
potential of the source. Consequently, electrons from the source electrode travel into
the 2DEG. The presence of electrons in the 2DEG screens the electric field induced
by the constant surface charge, thus the electron density is linearly increasing with
Vbg,th to values between 0.5 × 1011 cm−2 and 3 × 1011 cm−2 [128]. Irrespective of
undoping, the mobility after illumination at 1.6 K reaches a remarkably high value
of 5 × 106 cm2/Vs for n2D ∼ 3.5 × 1011 cm−2 , which is a record mobility value for
inverted GaAs/AlGaAs heterostructures [127]. Besides density controllability, these
devices enjoy the benefit of greatly reduced background impurity scattering. One of
134 CHAPTER 6. QUANTUM POINT CONTACT
(a)
(b)
Vbth
d2d1
2DEG
Φs0Φ i
Vbth
d2d1
2DEG
Φs0Φ i
Vb,th
2 DEG
Figure 6.2: (a) Growth structure of backgated 2DEG. d1 and d2 are the thickness ofGaAs and the two layers of AlGaAs and superlattice barriers respectively. (b) Banddiagram of backgated 2DEG operation. A thick solid line represents the case of Vb,th,a dotted line is for the above Vb,th and a thin solid line is for the below Vbg,th.
6.2. QUANTUM POINT CONTACT 135
120
100
80
60
40
20
0
V
H (
µV
)
1.00.80.60.40.20.0
B (T)
VB=2.2V
VB=2.3V
VB=2.4V
VB=2.5V
VB=2.7V
,BV n
3.0
2.8
2.6
2.4
2.2
D
en
sit
y (
10
15 m
-2)
2.72.62.52.42.32.2
VBG (V)
(a) (b)
Figure 6.3: (a) Measured Hall voltage as a function of external magnetic field perpen-dicular to 2DEG by changing the backgate voltages VBG. (b) The calculated electrondensity of 2DEG versus VBG.
these devices show density variations in measurements by applying a magnetic field
perpendicular to the 2DEG in Fig. 6.3.
6.2 Quantum Point Contact
Early attempts to form point contacts were tried with metals in a crude manner by
pressing two sharp ends [46,129]. It was extremely challenging by this method to fab-
ricate small confinements comparable to the Fermi wavelength λF around 1 A, not to
mention the lack of reproducibility. In these aspects, quantum point contacts (QPCs)
in semiconductors have superior benefits over metallic ones: relatively straightforward
width control around a few λF ∼ 40 nm in a high-mobility 2DEG and reproducible
fabrication recipe development. A QPC in a two-dimensional electron gas (2DEG)
system has been a prototypical device used to investigate low-dimensional mesoscopic
physics. Several methods to produce such constrictions are imprint lithography [130],
136 CHAPTER 6. QUANTUM POINT CONTACT
in-plane gate by focused ion etching [131], and split-gate technique [45]. The last
method is the most commonly used one based on the fact that negatively applied
voltages to lithographically patterned Schottky gates on top of a 2DEG creates an
electrostatic potential deep in a 2DEG. The electrostatic potential induces additional
spatial confinement to leave only one freely propagating direction (Fig. 6.4). It also
depletes electrons underneath to separate the electron reservoirs into two regions,
so-called source and drain. To adjust the effect of QPC potential in a controllable
manner with a Schottky gate voltage governs the width of the constriction (W ),
the route between source and drain, ultimately conductance. The QPC acts as an
electron waveguide, allowing certain modes to propagate through as W equals to mul-
tiples of half wavelength, W = nλ/2 with integer n. Note that a QPC is not truly
one-dimensional since W and length (L) of such confinements, in general, are rougly
λF ∼ W < L. It is rigorously quasi-one dimensional.
6.2.1 Conductance Quantization
As a manifestation of coherent electron propagation in a ballistic one-dimensional
electron waveguide, there are quantized conductance plateaus in integer-multiples
of the spin-degenerate quantum unit of conductance GQ = 2e2/h as the width of
a QPC increases. Recently, the quantum modes of coherent electrons under QPCs
were imaged by atomic force microscopy [132]. The simplest expression of current
in terms of GQ and the energy-independent transmission probabilities of transverse
channels Tn under finite drain-source voltages Vds is I =∑
nGQVdsTn. This simple
integer-plateau picture is true as long as Vds is kept small. A quantitative model to
produce such features and the non-zero Vds effects is presented and discussed in this
subsection.
A Saddle-Point Constriction
An electrostatic potential in a 2DEG associated with the negative gate voltage to the
Schottky electrodes is nicely modelled as a saddle-point potential, a linear combina-
tion of two harmonic oscillator potentials in a x−y plane. The saddle-point potential
6.2. QUANTUM POINT CONTACT 137
y
xZ
Vg < 0
y
xZ
y
x
y
xZ
Vg < 0
(a)
(b)
k
E
Figure 6.4: (a) Schottky-split techniques to form a QPC in a 2DEG sysgem. (b)Sketch of energy dispersion of electron reservoirs and QPC
138 CHAPTER 6. QUANTUM POINT CONTACT
under a zero magnetic field is written as
U(x, y) = U0 +1
2m∗ω2
yy2 − 1
2m∗ω2
xx2, (6.5)
assuming x is the longitudinal coordinate. U0, the potential at the saddle point has a
bias-voltage dependence, m∗ is the effective electron mass, and ωx, ωy are frequencies
of harmonic potentials. Transmission coefficients in the saddle-point constriction were
calculated analytically [133,134],
Tn(E) =1
1 + e−πǫn(E),
and ǫn, the lowest value of each transverse channel due to confinement potentials, is
found
ǫn(E) =2
~ωx
[
E − ~ωy(n+1
2− U0)
]
,
normalized by ~ωx. By definition, the current is computed from Tn(E) for EF −eVds >
0
I =2e
h
∫ EF
EF−eVds
∑
n
Tn(E).
Since Tn(E) is not a function of Vds, the differential conductance is dG = dI/dVds =
GQ
∑
n Tn(E). Note that ~ωx is taken as a energy scale throughout the theoretical
consideration. Figure 6.5 (a) depicts a saddle-point potential expressed in Eq. (6.5)
with the ratio of two frequencies ωy/ωx ≡ a = 2 and m∗ ∼ 0.067me for GaAs where
me is the electron mass. At first, the term U0 is assumed to be constant in Vds. In
the case of such a simple potential, the quantized energy level are readily expressed
ǫn(E) = 2
[
E − U0
~ωx
− ~ωy
~ωx
(n+1
2)
]
. (6.6)
In Eq. (6.6), the first term in the bracket is a variable corresponding to the voltage
applied to the Schottky gates on top of the 2DEG, eVg/~ωx. Defining z ≡ (E −
6.2. QUANTUM POINT CONTACT 139
U0)/~ωx, transmission coefficients are
Tn(z) =e2π[z−a(n+1/2)]
1 + e2π[z−a(n+1/2)].
Quantized plateaus in differential conductance are drawn in Fig. 6.5 (b) as Vg in-
creases, in other words, the constriction width becomes wider.
Non-Integer Conductance Plateaus
The previous saddle-point potential model is successful to describe quantized conduc-
tance plateaus at zero drain-source voltage. Natural questions arise that what would
happen as the drain-source voltage increases. Since transmission coefficients in this
model do not depend on the drain-source voltage, it is expected to have no drain-
source voltage effect on conductance trace. However, Kouwenhoven et al. reported
that nonlinear current-voltage characteristics were observed for high drain-source volt-
age opposed to the theoretical model prediction [135]. Based on experimental obser-
vation, the authors considered carefully the position of the Fermi level (EF ), chemical
potentials of left and right reservoirs, µL, µR in the energy dispersion diagram. As
long as the source-drain voltage is small enough, the number of transverse channels
is always the same for both forward and backward electron flow direction. However,
the story becomes different as soon as the difference of drain and source chemical
potentials is larger than the energy spacing of allowed bands. Figure 6.6 visualizes
the latter case in that µL lies in even below the lowest energy E0; thus, there is no
left moving channel. They claimed that the difference between forward and backward
propagating transverse channels within the bias energy window (µR −µL) would gen-
erate non-integer conductance plateaus for large Vds. A mathematical formulation
of this argument is quite straightforward. Suppose that the actual voltage drop at
left and right electrodes would be βeVds and (1 − β)eVds where 0 ≦ β ≦ 1. Then,
µR = EF + βeVds and µL = EF − (1 − β)eVds, which satisfies the initial construction
µR − µL = eVds.
A similar approach is taken to Eq. (6.5), introducing a drain-source voltage (Vds)
effect. Phenomenologically, the term U0 can be replaced by U0(Vds) = U0 − βeVds to
140 CHAPTER 6. QUANTUM POINT CONTACT
(b) 4
3
2
1
0
dG
(/
GQ
)
1086420 eVg (2π/hωx)
Vds = 0 V
xy
U
xy
U
xy
U
(a)
Figure 6.5: (a) The saddle-point potential for |ωy/ωx| = 2. (b) computed differentialconductance for the saddle-point potential (a).
6.2. QUANTUM POINT CONTACT 141
(1¡ ¯)eVds
¯eVds¹R
¹L
EF
E1
E0(1¡ ¯)eVds(1¡ ¯)eVds
¯eVds¯eVds¹R¹R
¹L¹L
EFEF
E1E1
E0E0
2.0
1.5
1.0
0.5
0.0
dG
(/
GQ
)
76543210eVg(2π/hωx)
Vds = 0 mV
Vds = 1 mV
Vds = 2 mV
(a)
(b)
Figure 6.6: (a) The actual voltage drop effect in left and right moving channels. (b)Computed differential conductance for non-zero Vds cases.
142 CHAPTER 6. QUANTUM POINT CONTACT
first order in Vds. β indicates how much portion of Vds drops across the source-device
interface and across the drain side, thus the range of β is [0,1]. U(Vds) yields Tn such
as
Tn(ǫn(E, Vds)) =eπǫn
1 + eπǫn, (6.7)
where ǫn(E, Vds) = 2~ωx
[
E − ~ωy(n+ 12) − U0(Vds)
]
.
The corresponding current for EF − eVds > 0 is
I(Vds) =2e
h
∫ µR
µL
Tn(ǫn(E, Vds))dE
=2e
h
∑
n
∫ µR
µL
eπǫn
1 + eπǫndE.
(6.8)
Substituting y = 1+ eπǫn = 1+ e2π
~ωx[E−~ωy(n+ 1
2)−U0(Vds)], the integral in E converts
into dy with dy = 2π/~ωxeπǫndE. This integral can be easily computed by simple
calculus:∫ y2
y1(~ωx/2π)(dy/y) = (~ωx/2π) ln(y2/y1). Equation (6.8) becomes
I(Vds) =2e
h
∑
n
~ωx
2πln
[
1 + e2π
~ωx[EF−~ωy(n+ 1
2)−U0(Vds)]
1 + e2π
~ωx[EF−eVds−~ωy(n+ 1
2)−U0(Vds)]
]
. (6.9)
Since transmission coefficients are a function of Vds as well as E, computing the
differential conductance from current requires extra care since both integrand and
integral ranges contain Vds. Suppose a case to differentiate with respect to x a integral
in y of a function f(x, y). If F (x, y) =∫
f(x, y), the remaining task is the following:
d
dx
∫ b(x)
a(x)
f(x, y)dy =d
dx[F (x, b(x)) − F (x, a(x))]
= [f(x, b(x)) − f(x, a(x)) + b′(x)f(x, b(x)) − a′(x)f(x, a(x))]
= b′(x)f(x, b(x)) − a′(x)f(x, a(x)) +
∫ b(x)
a(x)
d
dxf(x, y)dy.
Taking into account of the above derivative result, the differential conductance dG is
6.2. QUANTUM POINT CONTACT 143
derived as [136],
dG =2e2
h
∑
n
[
(
1 +dU0(Vds)
d(eVds)
)
e2π
~ωx[EF−eVds−~ωy(n+ 1
2)−U0(Vds)]
1 + e2π
~ωx[EF−eVds−~ωy(n+ 1
2)−U0(Vds)]
− dU0(Vds)
d(eVds)
e2π
~ωx[EF−~ωy(n+ 1
2)−U0(Vds)]
1 + e2π
~ωx[EF−~ωy(n+ 1
2)−U0(Vds)]
].
(6.10)
A natural choice of β is 0.5, which means that Vds drops symmetrically across drain
and source. Figure 6.6 (b) shows the differential conductance traces at three Vds
values, the evolution trend of integer plateaus to half-integer plateaus. In principle,
any non-integer conductance plateaus would emerge according to the value of β,
revealing the degree of symmetry between two reservoirs and a 2DEG.
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.30 -0.25 -0.20 -0.15 -0.10 VSchottky (V)
Date: 07-07-03
Vbg=2.5V400 uVpp -200uVoffset
Figure 6.7: The 0.7 structure from differential conductance measurements at 1.5 K.
144 CHAPTER 6. QUANTUM POINT CONTACT
6.2.2 The 0.7 Structure
In addition to conductance plateaus at integer multiples of quantum unit of conduc-
tance for low bias voltages across QPC, a remarkable feature around 0.6 - 0.8 GQ
has been identified [137]. It becomes denoted as the ‘0.7 structure’ or ‘0.7 anomaly’
since this feature is beyond simple non-interacting single particle picture. It is quite
robust and universal in a sense that any group observes this shoulder structure below
the first conductance plateau regardless of QPC shapes, 2DEG structures and het-
erostructure material. Not only QPC structures, quantum wires formed by cleaved
edge growth techniques preserve the feature as well [138]. Figure 6.7 plots a differen-
tial conductance of backgated quantum point contact at 1.5 K by sweeping Schottky
gate voltages, containing that deviating feature from the smooth steps starts near 0.7
GQ. Although its physical origin is under investigation and a microscopic-level un-
derstanding has yet been established, it is believed to be related to electron-electron
interactions unlike conductance plateaus, which are explained very well in terms of
non-interacting single electron approximation.
This speculation on correlated electrons was initiated from an experimental fact
that the 0.7 structure evolves to 0.5 when the spin degeneracy is lifted by in-plane
magnetic field [137] and enhanced g-factor under in-plane magnetic field [139], im-
plying that two-electron effect is reminiscent in a zero-field. In order to ferret out
such conjecture, many groups have been performing conductance measurement by
choosing practical tuning variables which may elucidate the behavior of interacting
electrons behavior in systems since late 1990s: channel length [140, 141], tempera-
ture [141–143], in-plane B-field [137, 143–146] bias Vds [142, 144, 145] and electron
density in 2DEG [140, 144, 147, 148]. Recently, more attentions move to shot noise
measurements near such structures [149,150].
There have been numerous theoretical models to explain the 0.7 structure in terms
of experimental variables such as temperature, in-plane magnetic field , bias Vds and
electron density: thermal activation field [151], Kondo effect [152], phenomenological
N-electron bound state model [153], spontaneous spin-polarization [154, 155] spin-
orbit coupling [156, 157], and phenomenological model [148]. In this chapter, the
spin-orbit coupling in the non-interacting single electron picture is discussed as a
6.2. QUANTUM POINT CONTACT 145
candidate to reproduce the 0.7 structure. This consideration lacks of many-body
effects, thus the further thought should be included in this approach; however, it
certainly has a pedagogical purpose.
Spin-Orbit Coupling
The spin-orbit (SO) coupling model takes a position of spontaneous spin polariza-
tion to a large extent. The formation of the 0.7 structure is caused by lifting spin
degeneracy due to spin-orbit interactions.
Spin-orbit coupling is pure quantum mechanical effect. It occurs when an electron
moves with a momentum ~p along the electric field ~E. Due to the relativistic motion,
the electron does feel the effective magnetic field ~B induced as ~B = −(~p × ~E)/mc
with an electron mass m and the speed of light c. Its Hamiltonian operator has a
form of HSO = ~S · ~L where ~S and ~L are spin angular momentum and linear angular
momentum operators. The spin-orbit (SO) Hamiltonian then is written in terms of
the momentum operator ~p, Pauli spin matrices ~σ and the gradient of a electrostatic
potential ~∇V ,
HSO =~
4m2c2(~∇V × ~p) · ~σ. (6.11)
In semiconductors, there are two types of SO interactions. These are related to
broken symmetries: Dresselhaus SO interaction is one due to bulk inversion asymme-
try described by the Hamiltonian HSO,D = β(σxkx−σyky) and Rashba SO interaction
is caused by structure inversion asymmetry described by the Rashba Hamiltonian
HSO,R = α(σxky −σykx) where α and β are coupling constants depending on material
characteristics. Between two SO interactions, for a QPC in a 2D plane, the Rashba
SO has bigger effect on transport properties. Therefore, here the Rashba SO effect is
focused on.
Case 1: Simple Harmonic Confinement Potential
Suppose the confinement potential in y-direction is U(y) = (1/2)~ω2, a simple har-
monic oscillator with an oscillating frequency ω. The Hamiltonian operator including
146 CHAPTER 6. QUANTUM POINT CONTACT
SO interaction becomes [156]
H =p2
2m∗ + U(y) − iασy∂
∂x+g∗µB
2~σ · ~B
=p2
2m∗ +1
2~ω2 − iασy
∂
∂x+g∗µB
2~σ · ~B,
(6.12)
under the influence of the magnetic field ~B. Herem∗, g∗ are effective electron mass and
effective g-factor and µB is the Bohr magneton. An ansatz solution of the Schrodinger
equation with Hamiltonian Eq. (6.12) has an easy form since x and y are separable,
ψ(x, y) = eikxφ(y)
(
ϕ↑
ϕ↓
)
, (6.13)
where k is the wavevector in x-direction, φ(y) is the wavefunction for the transverse
direction y and ϕ↑, ϕ↓ denote spinors with up and down spin configuration respec-
tively. Writing the Hamitonian operator in matrix using the basis of spinors is very
conveninent in order to calculate eigenenergies and eigenfunctions. Using the matri-
ces of spinors, ϕ↑ =
(
1
0
)
≡ | ↑〉 and ϕ↓ =
(
0
1
)
≡ | ↓〉, action of Pauli matrices
on the spinors yields:
σx| ↑〉 =
(
0 1
1 0
)(
1
0
)
=
(
0
1
)
= | ↓〉
σx| ↓〉 = | ↑〉
σy| ↑〉 =
(
0 −ii 0
)(
1
0
)
=
(
0
i
)
= i| ↓〉
σy| ↓〉 = −i| ↑〉〈↑ |σx| ↑〉 = 〈↓ |σx| ↓〉 = 〈↑ |σy| ↑〉 = 〈↓ |σy| ↓〉 = 0
〈↑ |σy| ↓〉 = −〈↓ |σy| ↑〉 = −i.
(6.14)
In the third term of Eq.(6.12) one should consider the eikx term of the wavefunction
6.2. QUANTUM POINT CONTACT 147
ψ due to the partial derivative with respect to x,
〈ψ(↑)| − iασy∂
∂x|ψ(↓)〉 = e−ikxφ∗(y)〈↑ | − iασy
∂
∂xeikxφ∗(y)| ↓〉
= −iα(ik)〈↑ |σy| ↓〉 = iαk
〈ψ(↓)| − iασy∂
∂x|ψ(↑)〉 = −iαk.
(6.15)
Together with Eqs.(6.14) and (6.15), the Hamiltonian matrix is
H =
(
~2k2
2m∗+ Etr
n iαk + g∗µB
2(Bx + iBy)
−iαk + g∗µB
2(Bx − iBy)
~2k2
2m∗+ Etr
n
)
, (6.16)
where Etrn is the quantized energy level from y-confinement equal to ~ω(n + 1/2).
Diagonalizing the Hamiltonian, the eigenenergies are
E±n =
~2k2
2m∗ + ~ω(n+1
2) ∓
[
(
g∗µBB
2
)2
+ 2µBαkB sin θ + (αk)2
] 1
2
. (6.17)
Since B2 = B2x + B2
y and By = B sin θ where θ is the angle along the y-direction,
Eq.(6.17) is expressed by the magnitude of total magnetic field, B and the angle θ.
SO - Effect in the Band Structure
Figure 6.8 compares two dispersion relations for zero (a) and non-zero (b) spin-
orbit coupling α by plotting Eq.(6.17) with dimensionless variables ξ(n) ≡ E±n /~ω,
P ≡ ~k/√
2m∗~ω and α′ ≡ α√
2m∗/~3ω. When magnetic field is absent, Eq.(6.17)
is much simplified to ξ(n) = P 2 ± α′P + (n + 1/2), which is used to generate plots
with α′ = 0 for (a) and α′ = 1 for (b) in Fig. 6.8. Note that non-zero magnetic field
induces anticrossings of channels which produce gaps in the spectra (Fig. 6.8 (c)).
The general formula for both non-zero B and θ is
ξ(n) = P 2 ∓√
B′2 + (α′P )2 + α′B′P sin θ + (n+1
2), (6.18)
where B′ is the normalized magnetic field such that B′ = g∗µB/2~ω.
148 CHAPTER 6. QUANTUM POINT CONTACT
P
(a)
(b)
P(c)
P
»
»
»
P
(a)
(b)
P(c)
P
»»»
»»»
»»»
Figure 6.8: (a) The band structure without spin-orbit interaction under zero magnetic
field ~B = 0 for the first five n. (b) The band structure with non-zero spin-orbit
interaction at ~B = 0. (c) The band structure with non-zero spin-orbit interactionat finite magnetic field. In all three cases magnetic field is perfectly aligned in x-direction, i.e. θ = 0. P and ξ are defined in the context.
6.2. QUANTUM POINT CONTACT 149
SO - Effect in Differential Conductance
The current from such band structures is written as
I =e
2π
∑
n,s
∫ ∞
−∞vns[Θ(vns)f(E, µL) + Θ(vns)f(E, µR)]dk, (6.19)
with the electron velocity vns for channel with band index n and spin s, step function
Θ and Fermi function of left and right reservoirs whose chemical potentials are µL
and µR respectively [156]. The velocity is the slope of the dispersion relation, vns =
(1/~)(∂En±/∂k), where
∂En±
∂k=
~2k2
m∗ ± 1
2
2α2k + αg∗µBB sin θ√
g∗µBB2
+ αg∗µBB sin θk + (αk)2
. (6.20)
Care should be taken to the integral range according to the positive and negative
velocities for forward and backward propagating modes in the band structures. The
interesting quantity now is the differential conductance obtained from taking a deriva-
tive of Eq.(6.18) with respect to the bias voltage eV = µL − µR. The explicit V
dependence appears in the Fermi functions . If µL = µR + eV , the surviving term is
only f(E, µL) = f(E, µR + eV ) after differentiation.
Assume that the magnetic field is aligned along x-direction. The differential con-
ductance is computed considering the velocity direction as well as temperature. The
results are plotted in Fig. 6.9 without and with the magnetic field for various tem-
peratures. All energy scales are renormalized by the confinement potential energy ~ω.
Note that the unit of differential conductance in this numerical analysis is taken to
be e2/h, the quantum unit of conductance without spin-degeneracy. When tempera-
ture is zero, the conductance trace for zero-magnetic field (Fig. 6.9 (a)) shows ideal
step-like plateaus, indicating that the Fermi functions are exactly step functions as
expected. In addition, the number of propagating channels in both directions remains
the same. However, as soon as the magnetic field is turned on, due to the presence
of anticrossings in the band structure, there exist dips which correspond to energy
150 CHAPTER 6. QUANTUM POINT CONTACT
¹¹
dG(=(e2 h))
dG(=(e2 h))
dG(=(e2 h))
dG(=(e2 h))
¹¹
(a)
(b)
Figure 6.9: Computed differential conductance as spin-orbit coupling in a simpleharmonic oscillator potential is on while the magnetic field is kept zero in (a) and themagnetic field is 0.05 in the unit of g∗µBB/2~ω in (b).
6.2. QUANTUM POINT CONTACT 151
gaps among bands, making the participant channel numbers different. The trace at
zero temperature evolves smoothly to the ones at finite temperature, resembling the
observed 0.7 structure before conductance plateaus are completely washed out due to
thermal broadening. The key to validate this model by experiments is to choose the
appropriate condition to satisfy the numerical energy scaling ranges. Up to now, the
electrostatic potential formed in the 2DEG to confine systems is not well controllable
practicallyt; however, other parameters would be rather flexible to be selected for
tests.
dG(=(e
2 h))
dG(=(e
2 h))
¹¹
Figure 6.10: Computed differential conductance as spin-orbit coupling is on whilemagnetic field is kept zero.
Case 2: Saddle-Point Potential
Although the simple harmonic potential produces a promising perspective towards
the 0.7 structure, it is a natural extension to apply the spin-orbit concept to the
saddle-point potential, which is rather physical model as a confined potential in real
2DEGs. The simplest attempt is done for the potential keeping only the linear Vds
152 CHAPTER 6. QUANTUM POINT CONTACT
term and for spin-orbit coupling without a magnetic field applied in the system. The
absence of the magnetic field does not induce anticrossings, consequently no bumps
below plateaus for the saddle-point potential shown in Fig. 6.10. Therefore, in this
story there should be non-zero magnetic field to cause anticrossings between bands.
The tasks become quite complicated to compute the differential conductance with
nontrivial band structures arising from a nonzero magnetic field and its orientation
in the system. However, these remaining tasks may be worthy of being pursued
in future, examining the effect of combining the potential shape and the spin-orbit
interaction as one of the approaches regarding the microscopic understanding of the
0.7 structure.
The initial attempt to incorporate the spin-orbit interactions in the single-electron
picture with a harmonic oscillator seems promising to produce a similar shape of the
0.7 structure near the 0.7 GQ value. This interaction differentiates spin-up and spin-
down electrons which propagate through the narrow channel. This concept of this
model is reasonable and plausible in the real system, thus the further rigorous calcu-
lation including the saddle-point potential and careful experimental implementation
would be highly on demand.
6.3 Differential Conductance
Implementing experimental methodology described in Chapter 4, nonequilibrium
transport of a single QPC through differential conductance and low-frequency shot
noise were studied at 1.5 K. Figure 6.11 shows the shape of a Schottky split-gate on top
of a 2DEG taken by a scanning electron microscope and the image of a wire-bonded
whole chip which has the Hall-bar structure with Ohmic contacts and voltage probes.
The Hall-bar structure has a capability to characterize the quality of the 2DEG as
well as to access quantum Hall regimes by applying a perpendicular magnetic field
to the 2DEG plane. Backgated QPCs in question have several ways to regulate the
electron density. Three experimental variables are used:(1) backgate voltage (Vbg) to
vary the electron sheet density in the 2DEG, (2) Schottky split-gate voltage (Vg) to
control the entrance of electrons from reservoirs, and (3) drain-source voltage (Vds)
6.3. DIFFERENTIAL CONDUCTANCE 153
(a)
(b)
(a)
(b)
Figure 6.11: (a) Scanning electron microscope (SEM) image of a quantum pointcontact in a AlGaAs/GaAs 2DEG. (b) The Hall-bar structure of the wirebondeddevice taken by SEM.
154 CHAPTER 6. QUANTUM POINT CONTACT
-0.30
-0.25
-0.20
-0.15
-0.10
Vg (
V)
3210-1-2-3
Vds (mV)
(b)
1.00.9 0.9
0.50.5
2.5
2.0
1.5
1.0
0.5
dG (
/GQ
)
-3 -2 -1 0 1 2 3
Vds (mV)
(a)
1.5
1.0
0.5
dG
(/G
Q)
-0.32 -0.28 -0.24 -0.20
Vg (V)
Vds = 0.0 mV = 0.7 mV
= 1.0 mV
= 1.5 mV
= 2.0 mV = 2.5 mV
(c)
Figure 6.12: (a) Experimental differential conductance dG by a sweep of Vg at fixedVbg = 2.3 V at finite Vds (b) Transconductance dG/dVg (c)Vds dependence
6.3. DIFFERENTIAL CONDUCTANCE 155
to determine number of electrons to flow through the potential constriction. This
section discusses experimental data in terms of three knobs, Vds, Vg and Vbg.
6.3.1 Non-integer Conductance Plateaus at Finite Bias Volt-
age
The Hall-bar structure allowed us to measure four-probe measurements, taking the
actual voltage drop across the QPC. At first, we fixed a 2DEG electron density by
applying a certain backgate voltage Vbg. The Vbg was chosed at 2.3 V for Fig. 6.12.
The measured dG with an ac bias voltage Vac ∼ 100 µV is plotted as a function
of the drain-source voltage Vds and the split-gate voltage Vg. The values of dG are
normalized by the spin-degenerate quantum unit of conductance GQ. Dark regions in
Fig. 6.12(a) emerge as dG traces converge according to Vg change. They correspond to
the conductance plateaus. As Patel and his colleagues pointed out, dG flattens around
integer-multiples of GQ along Vds ∼ 0, whereas away from Vds ∼ 0 dG approaches
plateaus but at different positions [139]. The non-integer conductance plateaus at
different Vds values are clearly illustrated by plotting individual graphs in Fig. 6.12
(c), where the first step emerges below 0.5 GQ when Vds = - 2.5 mV.
We compute the transconductance d(dG)/dVg by differentiating dG in terms of
Vg as a post-analysis. The quantity is presented in a two-dimensional image graph
(Fig. 6.12(b)). Here, black areas correspond the plateaus due to the small difference
between traces along Vg axis. Inside the first big diamond black area, a V-shape red
structure exists. It recognizes the 0.9 structure from the GQ plateau. Furthermore,
we notice that the transition behavior is not identical over the whole conductance
values for finite Vds. Below GQ, an additional shoulder structure around 0.7 GQ is
manifest and it moves to 0.9 GQ. Then the plateau clearly forms below 0.5 GQ at
a large Vds. In contrast, above GQ, as Vds increases, no structure similar to the 0.7
anomaly is apparent and the plateau shows an increasing manner. The appearance of
the non-integer conductance plateaus in terms of Vds is understood quantitatively by a
Vds-dependent saddle-point potential model in x and y directions described previously
upto second-order Vds terms U0(Vds) = U0 − βeVds + γeV 2ds/2 [136]. A higher-order
156 CHAPTER 6. QUANTUM POINT CONTACT
2.5
2.0
1.5
1.0
0.5
dG
(/
GQ
)
-3 -2 -1 0 1 2 3
Vds (mV)
(a) (b)
(d)(c)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
2.5
2.0
1.5
1.0
0.5
dG
(/
GQ
)
-3 -2 -1 0 1 2 3
Vds (mV)
(a) (b)
(d)(c)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
dG( =GQ)
Figure 6.13: (a) Observed differential conductance traces at each Schottky gate volt-age as a function of Vds. All date were taken at Vbg = 2.3V and 1.5 K. (b) Computeddifferential conductance with the Vds-dependent saddle-point potential up to the lin-ear term, i.e. γ = 0. Including second-order corrections in Vds with two oppositesigns of the coefficient, (c) γ > 0 and (d) γ < 0.
6.3. DIFFERENTIAL CONDUCTANCE 157
term would exhibit some aspects of the experiment. It indicates that γ relates to the
trend of plateau movements for finite Vds values: A negative quadratic term yields a
decreasing pattern while a positive term yields an increasing pattern shown in Fig.
6.12(c) and (d). The contribution of the theoretical model is to provide qualitative
picture of plateau evolution along with Vds; however, the model fails to replicate the
abnormal plateaus, the 0.7 GQ and the 0.9 GQ, which is beyond the present theoret-
ical model. As suggested in the previous spin-orbit coupling section, one immediate
action is to incorporate the spin-orbit coupling and the second-order saddle-point
potential in order to see whether this action provides promising perspectives or not.
Otherwise, the next candidate is to approach the whole system in the consideration
of many-body interactions.
6.3.2 Density Effect
5
4
3
2
1
G (
/G
Q)
-0.6 -0.5 -0.4 -0.3 -0.2 Vg (V)
5
4
3
2
1
G
(/
GQ
)
-0.6 -0.5 -0.4 -0.3 -0.2 Vg (V)
(a) (b)
Figure 6.14: The tuning variables are Vg and Vbg and dG is measured at four-probetechniques with ac signal Vac ∼ 50 µV and the dc bias (a) Vds = 0 mV and (b) Vds =2 mV. The Vbg varies from 3.0 V (rightmost) to 2.58 V (leftmost) by 0.01V interval.
One of the aim to fabricate backgated QPCs is to study the electron density effect
on the 0.7 anomaly. This time the constant variable throughout measurements is the
158 CHAPTER 6. QUANTUM POINT CONTACT
drain-source voltages and the sweeping variables are Vg and Vbg.
Figure 6.14 compares two cases of different dc drain-source voltages. The common
feature regardless of Vds sizes is that the threshold value of Vg at which electrons start
flow through the constriction moves leftward as larger Vbg is applied. Vbg indeed
controlls the electron density in a 2DEG. The leftward motion of Vg means that the
more electrons reside in the 2DEG at large Vbg, the bigger pinch-off voltage is required
to deplete all 2DEG electrons. There is rather weak structure near the 0.7 GQ, and no
significant evolving trend of the 0.7 structure is captured as a function of the electron
density or Vbg. However, as the dc bias voltage increases in (b), the robust plateaus
below GQ emerge near 0.5 GQ as well as integer-GQ plateaus. Nuttinck and NTT
colleagues reported a vivid trend about the 0.7 structure evolution into the 0.5 GQ
with the same device structures in the limit of low electron density, claiming that two
degenerate channels are splitting and a two-channel model in the Landauer - Buttiker
formalism may explain it in a phenomenological level [147]. The failure to reproduce
their results would be conjectured that the density is not significantly varying from
the initial and the final Vbg. Thus, the future work should be scrutinizing the role
of electron density in an improved setup, resolving the anomaly structures and the
emergence of integer plateaus together with 0.5 GQ steps.
6.4 Low-frequency Shot Noise
The early shot noise measurements of a single QPC have revealed important in-
formation of ballistic quantum transport. The ballistic transport behaviors related
the conductance plateaus obtained from the shot noise measurements have been
well interpreted in terms of noise suppression within the Landauer-Buttiker formal-
ism [158]. Recent efforts of shot noise measurements have focused the 0.7 anomaly
structures [149, 150, 159]. These work have reported complete shot noise suppres-
sion [159] and a partial suppression in terms of Fano factor whose number is in the
range predicted by a two-channel model [149]. New shot noise measurements of the
0.7 structure provide further evidence that the evolution of noise behavior from the
point of 0.7 GQ to the symmetric point 0.5 GQ occurs as the in-plane magnetic field is
6.4. LOW-FREQUENCY SHOT NOISE 159
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20Vg (V)
25x10-6
20
15
10
5
Un
no
rma
lize N
ois
e (a
.u.)
Vds = 0.7 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G
(/
GQ
)
-0.35 -0.30 -0.25 -0.20 -0.15 V g (V)
Vds = - 0.7 mV
= - 1.0 mV
= - 1.5 mV
= - 2.0 mV
= - 2.5 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)
100x10-6
80
60
40
20
0
Un
no
rm
aliz
e N
ois
e (a
.u.)
Vds = 2 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)
80x10-6
60
40
20
0
Un
no
rm
aliz
e N
ois
e (a
.u.)
Vds = 2.5 mV
(a) (b)
(c) (d)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20Vg (V)
25x10-6
20
15
10
5
Un
no
rma
lize N
ois
e (a
.u.)
Vds = 0.7 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G
(/
GQ
)
-0.35 -0.30 -0.25 -0.20 -0.15 V g (V)
Vds = - 0.7 mV
= - 1.0 mV
= - 1.5 mV
= - 2.0 mV
= - 2.5 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)
100x10-6
80
60
40
20
0
Un
no
rm
aliz
e N
ois
e (a
.u.)
Vds = 2 mV
3.0
2.5
2.0
1.5
1.0
0.5
0.0
G (
/G
Q)
-0.35 -0.30 -0.25 -0.20 -0.15Vg (V)
80x10-6
60
40
20
0
Un
no
rm
aliz
e N
ois
e (a
.u.)
Vds = 2.5 mV
(a) (b)
(c) (d)
Figure 6.15: Bias dependence at Vbg = 2.3 V. (a) Vds = 0.7 mV. (b) Vds = 2 mV. (c)Vds = 2.5 mV. (d) Vds dependent conductance.
160 CHAPTER 6. QUANTUM POINT CONTACT
applied although the paper does not explicitly state the value of the Fano factor [150].
Using the quantitative information in the paper [150], the Fano factor is managed to
be estimated. The estimate says that the Fano factor is around 0.18 near the 0.7
structure. Thus it supports again that repulsive correlation among electrons near
the structure may be involved; however, these experimental data are yet to solve the
puzzle of the 0.7 structure origin. This section discusses rather conceptually trans-
parent aspects of noise properties based on experimental results in conjunction with
differential conductance.
6.4.1 Noise Suppression at Non-integer Conductance Plateaus
Together with the differential conductance, two-terminal shot noise measurements
were executed at 1.5 K with all necessary technical strategies described in the Chapter
4. The tank circuit resonance occurs around 12 MHz at 1.5 K and a bandpass filter
on the output signal line from the device after the cryogenic amplifier comprise of a
21.4 MHz low-pass filter and a 5.6 MHz high-pass filter. One difference in the setup
from the experimental setup of carbon nanotubes presented in Chapter 5 is that there
is no full-shot noise source in parallel with a QPC device. Therefore, we have not
yet succeeded in absolute calibration of measurement apparatus; however, we are still
able to extract interesting and reproducible trend. For the shot noise signal, there
is the upper bound of Vds to beat the background noise related to the signal-to-the
noise ratio. This is is the limit of the present circuit and the drain-source voltage
should be more than 500 µV.
Figure 6.15 (a)-(c) show three representative plots of shot noise along with linear
conductance versus Vg at three different Vds in Fig. 6.15 (d). Similar behaviors are
observed in other devices as well. No matter what values of Vds are applied, the shot
noise level is clearly minimal when conductance G = I/Vds reached about GQ and
2 GQ regardless of calibration. The degree of the suppression at 3 GQ becomes less
significant for a large Vds. In the transient zones between the multiples ofGQ, the noise
characteristic is rather complex. Below the first plateau, the noise suppression appears
around 0.6 GQ and 0.9 GQ until Vds ∼ 1.5 mV (Fig. 6.15(a)). As Vds further increases,
6.4. LOW-FREQUENCY SHOT NOISE 161
-0.35
-0.30
-0.25
-0.20
-0.15
V
g (
V)
-3.0 -2.0 -1.0
Vds (mV)
(a) (b)
-0.35
-0.30
-0.25
-0.20
-0.15
V
g (
V)
-3.0 -2.0 -1.0
Vds (mV)
2.5
2
1.5
1
0.9
0.7 0.5
2 GQ
1 GQ
1.5 GQ
0.5 GQ
-0.35
-0.30
-0.25
-0.20
-0.15
V
g (
V)
-3.0 -2.0 -1.0
Vds (mV)
(a) (b)
-0.35
-0.30
-0.25
-0.20
-0.15
V
g (
V)
-3.0 -2.0 -1.0
Vds (mV)
2.5
2
1.5
1
0.9
0.7 0.5
2 GQ
1 GQ
1.5 GQ
0.5 GQ
2 GQ
1 GQ
1.5 GQ
0.5 GQ
Figure 6.16: (a) Two-dimensional plot of shot noise raw data at Vbg = 2.3 V withlines at the conductance values. (b) The contour plot of conductance values in theunit of GQ simultaneously taken with shot noise measurements.
these locations move down to 0.5 GQ and 0.8 GQ (Fig. 6.15(b)). And eventually the
suppressed noise is found only at 0.4 GQ for Vds > 2.5 mV (Fig. 6.15(c)). In contrast,
when G is higher than GQ, only one additional noise reduction is found about 1.6 GQ
or 1.7 GQ regardless of the magnitude of Vds. The plateau structures in G gradually
washes out as Vds increases as shown in Fig. 6.15(d). Although the conductance loses
its step-behavior in high Vds, the noise suppression appears in a clear and robust
manner.
The two-dimensional image plot of shot noise as a function of Vg and Vds is of
ease to perceive the continuous shot noise behavior. The black color depicts the base
shot noise level. The occurrence of the suppressed shot noise can be easily seen in
units of GQ. Furthermore, the actual plot contains other noticeable features. The left
162 CHAPTER 6. QUANTUM POINT CONTACT
figure has eye-guide lines for conductance values taken simultaneously along the dc
line. The colored contour plot of conductance G = I/Vds(Fig. 6.16(b)) helps to see
the relation of G and the shot noise. Again under GQ, several black strips are visible:
The upper strip relates to the shot noise suppression around GQ, and the lower two
ones start at the conductance values 0.7 GQ and 0.9 GQ. For a large Vds, the shot
noise suppression occurs at less than 0.5 GQ. The shot noise signal in higher G has
a rather simple pattern: the reduced noises are observed around 1.6 or 1.7 GQ and 2
GQ as previously stated.
We notice that the shot noise behavior in the transient zone between the integer
multiples of GQ shares some features with the transconductance two-dimensional
image plot (Fig. 6.12(b)). The peaks in the transconductance correspond to the
larger shot noise signals and the dark areas in the transconductance match to the
black strips in the shot noise image. Moreover, both the transconductance and the
shot noise share common features for G < GQ; the 0.7 structure can be distinctive
and the location of the noise suppression and the new plateaus in d(dG)/dVg occur
around 0.4 GQ as Vds > 2 mV. Within the saddle-point potential model, d(dG)/dVg
is expressed in terms of Ti(1 − Ti) where Ti is the i-th one-dimensional (1D) channel
transmission probability. Since the shot noise has a term of Ti(1−Ti) for a small energy
window, two quantities are closely related. It is not, however, obvious to predict the
response of the shot noise for a large Vds because the shot noise is obtained from the
integral of the energy dependent transmission probability. Qualitatively, the noise
suppression around the plateaus can be expected based on the fact that the current
fluctuations can be zero or low when the current remains constant.
The different characteristics in both the transconductance and the shot noise
are observed in the region of G < GQ and G > GQ. This observation is certainly
beyond the simple saddle-point potential model in a single-particle approximation. In
particular, it is surprising to have the strongly suppressed shot noise at 0.7 GQ. This
empirical fact means that electrons are regulated at 0.7 GQ by a certain governing
physical mechanism. The possible factor relating to the mechanism of the 0.7 anomaly
would be the density of electrons. The shot noise study in terms of the electron density
would provide more information to explore this question in the future.
6.4. LOW-FREQUENCY SHOT NOISE 163
3.0
2.8
2.6
2.4
V
bg (
V)
-0.6 -0.5 -0.4 -0.3 -0.2
Vg (V)
4
3.6
3
2.6
2
1.5
1
0.9
500
400
300
200
100
0
x1
0-6
3.0
2.9
2.8
2.7
Vb
g (V
)
-0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30
Vg (V)
3.6
3
.5
3
3
2.6
2.5
2
2
1.5
1
1 0
.5
100
80
60
40
20
0
x1
0-6
(a)
(b)
3.0
2.8
2.6
2.4
V
bg (
V)
-0.6 -0.5 -0.4 -0.3 -0.2
Vg (V)
4
3.6
3
2.6
2
1.5
1
0.9
500
400
300
200
100
0
x1
0-6
3.0
2.9
2.8
2.7
Vb
g (V
)
-0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30
Vg (V)
3.6
3
.5
3
3
2.6
2.5
2
2
1.5
1
1 0
.5
100
80
60
40
20
0
x1
0-6
(a)
(b)
Figure 6.17: Two-dimensional image plot of shot noise versus Vg and Vbg at (a)Vds = 1 mV. (b) Vds = 2 mV. The straight lines with number are corresponding tothe conductance values normalized by GQ.
164 CHAPTER 6. QUANTUM POINT CONTACT
6.4.2 Density Effect
Similar to the differential conductance, density effect on the shot noise has been
studied in the preliminary level. Again, the sweeping variables are Vg and Vbg while
Vds is fixed at a certain value. Two-dimensional image plots of the shot noise signals in
Fig. ?? exhibit the consistent features of the differential conductance: the threshold Vg
appears at higher values for a high 2DEG density; and the suppressed noise at 0.9GQ
evolves to 0.5GQ as Vds increases. Unfortunately, the 0.7 structure is not recognizable
in these plots although individual noise trace contains a partial or full suppression
discussed in the previous subsection. The better resolution of the experiment is
required by improving technical side.
6.5 Summary
The main theme of Chapter 6 is to investigate quantum ballistic transport properties
of a quantum point contact in a GaAs/AlGaAs heterostructure. The general char-
acteristics of two dimensional electron gas systems as a mother material of a QPC
have been discussed in terms of energy band diagrams followed by the formation of
a QPC, introducing the spatial confinement potential models. The transport quanti-
ties, current and conductance are computed in the non-interacting Landauer-Buttiker
formalism. Chapter 6 has also discussed the unresolved feature so-called the 0.7 struc-
ture, and as a theoretical attempt to understand the origin of the 0.7 structure, the
effect of the spin-orbit interaction has been studied. In the last two sections, exper-
imental results of differential conductance and shot noise measurements performed
at 1.5 K are presented and the observations are analyzed. Two independent mea-
surements are closely tightened such as the suppressed noise values are founded near
the conductance plateaus. Therefore, further measurements on both quantities are
essential to explore transport properties including the 0.7 structure in future.
Chapter 7
Conclusions
If I can stop one heart from breaking,
I shall not live in vain;
If I can ease one life the aching,
or cool one pain,
or help one fainting robin
Onto his nest
I shall not live in vain.
− Emily Dickinson
Mesoscopic systems become essential to explore the transition between classical
physics and quantum physics for last several decades. The continuing fabrication
advancement has expanded the coverage of regime and the diversity of structures.
Indeed, unique physics phenomena have been revealed in designed systems, validating
the postulates of quantum physics and providing the new insights of perspective
prototypical devices in the areas of information storage, information transportation,
and information manipulation.
Among numerous specific systems, this thesis has put particular attention to two
one-dimensional mesoscopic structures, investigating the electron motions in non-
equilibrium condition at cryogenic temperatures. It has inquired governing principles
165
166 CHAPTER 7. CONCLUSIONS
to control the behavior of charge carriers given situation. It has described theoreti-
cal aspects of many-electron systems and applied such knowledge into experimental
observations, attempting to understand empirical facts as much as possible and pos-
tulating hypotheses for further investigation.
The study of single-walled carbon nanotubes (SWNTs) has raised interesting is-
sues to probe intrinsic electron-electron interactions which are unavoidable in one-
dimensional systems. The electrical measurements of differential conductance and
shot noise exhibited the signatures of strong interaction among electrons, which
are qualitatively and quantitatively explained in Tomonaga-Luttinger liquid (TLL)
model. Still some features appearing in data are beyond the TLL model, requiring
modified assumptions and better modelling of systems in question. The study can
claim that our experimental techniques provide a way to quantify the contributions
of transmitted and backscattered currents in a three-terminal device which couples
well to the electron reservoirs. The natural extension is to investigate semiconducting
SWNTs, multi-walled carbon nanotubes, and single metallic quantum wire in terms
of electron-electron interactions. It can apply to understand spintronic devices with
ferromagnetic reservoirs with considerations.
The study of quantum point contact has also explored low-dimensional physics,
mainly focusing on the effect of physical variables to regulate electron density. The
new contribution is the detailed shot noise measurement complimentary to differential
conductance. They will be of importance to seek the origin of the universal system
‘0.7 structure’. At present, there are many attempts to explain the feature, but the
complete picture has yet to come. The thesis gives a try to analyze the single-particle
picture with spin-orbit coupling for the 0.7 structure although it is mostly considered
as the symptom of many-body effect. In order to identify the microscopic level un-
derstanding of such feature, careful experiments should be designed and performed
together with advanced theoretical models.
This thesis is along the journey to explore novel physical phenomena occurring
in mesoscopic structures. It is based on previous understanding on them and it is
hoped to add even a slight knowledge in mesoscopic field in order to attain the truth
of science and nature.
Appendix A
Physical Constants
Symbols Physical Constants MKS unit CGS unit eV unit
h Planck's constant 6.62618 £10¡34 J-s 6.62618 £10¡27 erg-s~ h/2¼ 1.05459 £10¡34 J-s 1.05459 £10¡27 erg-s 6.58217 £10¡16 eV-se elementary charge 1.6 £10¡19 C 4.803 £10¡10 esuc speed of light 2.99792 £108 m/s 2.99792 £1010 cm/s® ¯ne structure constant = e2/hc 1/137.036me electron mass 9.10953 £10¡31 kg 9.10953 £10¡28 g 0.511 MeV/c2
mh proton mass 1.67265 £10¡27 kg 1.67265 £10¡24 g 938.279 MeV/c2
aB Bohr radius 0.52918 ºA 0.52918 £10¡8 cm¹B Bohr magneton = e~/2mec 0.9273 £10¡27 J/gauss 0.9273 £10¡34 erg/gauss 5.78838 £10¡9 eV/gausskB Boltzmann's constant 1.3806 £10¡23 J/K 1.3806 £10¡30 erg/KR Gas constant 8.3145 J-K/molNA Avogadro's number 6.02205 £1023 / mol
Symbols Physical Constants MKS unit CGS unit eV unit
h Planck's constant 6.62618 £10¡34 J-s 6.62618 £10¡27 erg-s~ h/2¼ 1.05459 £10¡34 J-s 1.05459 £10¡27 erg-s 6.58217 £10¡16 eV-se elementary charge 1.6 £10¡19 C 4.803 £10¡10 esuc speed of light 2.99792 £108 m/s 2.99792 £1010 cm/s® ¯ne structure constant = e2/hc 1/137.036me electron mass 9.10953 £10¡31 kg 9.10953 £10¡28 g 0.511 MeV/c2
mh proton mass 1.67265 £10¡27 kg 1.67265 £10¡24 g 938.279 MeV/c2
aB Bohr radius 0.52918 ºA 0.52918 £10¡8 cm¹B Bohr magneton = e~/2mec 0.9273 £10¡27 J/gauss 0.9273 £10¡34 erg/gauss 5.78838 £10¡9 eV/gausskB Boltzmann's constant 1.3806 £10¡23 J/K 1.3806 £10¡30 erg/KR Gas constant 8.3145 J-K/molNA Avogadro's number 6.02205 £1023 / mol
167
Appendix B
Conversion Tables
hν = kBT,
ν =c
λ,
ν =1
time.
(B.1)
B.1 Energy and temperature
temperature energy
1 K 86 µeV
4 K 0.344 meV
300 K 25 meV
168
B.2. FREQUENCY, TEMPERATURE, ENERGY, WAVELENGTH AND TIME169
B.2 Frequency, temperature, energy, wavelength
and time
frequency temperature energy wavelength time
10 MHz 500 µK 40 neV 30 m 100 ns
1 GHz 50 mK 4 µeV 30 cm 1 ns
21 GHz 1 K 90 µeV 14 mm 50 ps
240 GHz 12 K 1 meV 1.24 mm 4 ps
360 THz 17400 K 1.5 eV 824 nm 3 fs
Rule of Thumbs
• Temperature vs. Energy : 1 K ∼ 80µ eV
• Wavelength vs. Energy : 1 nm ∼ 2 meV at λ ∼ 780 nm
Wavelength vs. Frequency : 1 nm ∼ 500 GHz at λ ∼ 780 nm
• LC circuit resonant frequeny: 1 nH + 1 pF ∼ 5 GHz, 1 µ H + 1 nF = 5 MHz
Appendix C
Statistics of Particles
The distribution function is a statistical concept, which provides the information
about the probability of a particle in energy state E.
C.1 The Maxwell-Boltzmann Distribution
Identical but distinguishable (classical) particles such obey the Maxwell-Bolzmann
(MB) distribution at given temperature T and at energy E
fMB = e−(E−µ)/kBT .
As an example, for a particle whose mass is m with a velocity ~v, such molecular
distribution is specified as
f~v = 4π
(
m
2πkBT
)3/2
~v2exp
[−m(~v)2
2kBT
]
,
from which many fundamental gas properties can be explained.
170
C.2. THE FERMI-DIRAC DISTRIBUTION 171
C.2 The Fermi-Dirac Distribution
Identical indistinguishable particles with half-integer spin obey the Fermi-Dirac (FD)
distribution, fulfilling the Pauli’s exclusion principle in the quantum world. It is
essential to study electrons in matels and conduction processes in semiconductors.
fFD(E, µ, T ) =1
e(E−µ)/kBT + 1,
where µ is the chemical potential.
The FD distribution approaches to the MB distribution in the limit of high tem-
perature and low density.
Integrals
(i)∫∞0fi(E)dE
Substituting x = e(E−µi)/kBΘ, kBΘdx = xdE and integrands come from x =
e−µi/kBΘ ≪ 1 ( µ1 ≫ kBΘ) to x→ ∞.
∫ ∞
0
1
e(E−µi)/kBΘ + 1dE ≈ kBΘ
∫ ∞
0
(
1
x+ 1
)(
1
x
)
dx
= kBΘ
∫ ∞
0
(
1
x− 1
x+ 1
)
dx
≈ kBΘ
[
ln
(
x
1 + x
)]∞
e−µi/kBΘ
≈ kBΘ(0 − ln(e−µi/kBΘ)) ∼ µi.
(ii)∫∞0fi(E)(1 − fi(E))dE
172 APPENDIX C. STATISTICS OF PARTICLES
Similarly, put x = e(E−µi)/kBΘ.
∫ ∞
0
fi(E)(1 − fi(E))dE =
∫ ∞
0
(
1
e(E−µi)/kBΘ + 1
)(
e(E−µi)/kBΘ
e(E−µi)/kBΘ + 1
)
dE
=
∫ ∞
e−µ/kBΘ
kBΘ1
(x+ 1)2dx
= kBΘ
[
− 1
x+ 1
]∞
e−µi/kBΘ
≈ kBΘ
(
1
1 + e−µi/kBΘ
)
≈ kBΘ.
(iii)∫∞0fi(E)(1 − fj(E))dE
Again, set x = e(E−µi)/kBΘ and a = e(µi−µj)/kBΘ
∫ ∞
0
fi(E)(1 − fj(E))dE =
∫ ∞
0
(
1
e(E−µi)/kBΘ + 1
)(
e(E−µj)/kBΘ
e(E−µj)/kBΘ + 1
)
dE
=
∫ ∞
e−µ/kBΘ
kBΘ
(
1
x+ 1
)(
a
ax+ 1
)
dx
= kBΘa
1 − a
∫ ∞
e−µ/kBΘ
(
1
x+ 1− a
ax+ 1
)
dx
= kBΘa
1 − a
[
ln
(
x+ 1
ax+ 1
)]∞
e−µi/kBΘ
≈ kBΘa
1 − aln
∣
∣
∣
∣
1
a
∣
∣
∣
∣
≈ (µi − µj)
(
e(µi−µj)/kBΘ
e(µi−µj)/kBΘ − 1
)
.
Interchanging i and j and doing some algebra,
∫ ∞
0
fj(E)(1 − fi(E))dE ≈ (µi − µj)
(
1
e(µi−µj)/kBΘ − 1
)
.
C.2. THE FERMI-DIRAC DISTRIBUTION 173
Thus,
∫ ∞
0
fi(E) [1 − fj(E)] + fj(E) [1 − fi(E)]dE
≈ (µi − µj)
(
e(µi−µj)/kBΘ + 1
e(µi−µj)/kBΘ − 1
)
= (µi − µj) coth
(
µi − µj
2kBΘ
)
.
(iv)∫∞0fi(E)(1 − fj(E + C))dE where C is a constant
Since C is a constant, modifying the above ‘a’ provides the answer. Let b =
e(C+µi−µj)/kBΘ with the same ‘x’,
∫ ∞
0
fi(E) [1 − fj(E + C)] dE
=
∫ ∞
0
(
1
e(E−µi)/kBΘ + 1
)(
e(E+C−µj)/kBΘ
e(E+C−µj)/kBΘ + 1
)
dE
=
∫ ∞
e−µ/kBΘ
kBΘ
(
1
x+ 1
)(
b
bx+ 1
)
dx
≈ (C + µi − µj)
(
e(C+µi−µj)/kBΘ
e(C+µi−µj)/kBΘ − 1
)
.
(v)∫∞0fj(E + C) [1 − fi(E)] dE where C is a constant
With b = e(C+µi−µj)/kBΘ,
∫ ∞
0
fj(E + C) [1 − fi(E))] dE
=
∫ ∞
0
(
1
be(E−µi)/kBΘ + 1
)(
e(E+C−µi)/kBΘ
e(E+C−µi)/kBΘ + 1
)
dE
≈ (C + µi − µj)
(
1
e(C+µi−µj)/kBΘ − 1
)
.
174 APPENDIX C. STATISTICS OF PARTICLES
Combining (iv) and (v),
∫ ∞
0
fi(E) [1 − fj(E + C)] + fj(E + C) [1 − fi(E)]dE
≈ (C + µi − µj)
(
e(C+µi−µj)/kBΘ + 1
e(C+µi−µj)/kBΘ − 1
)
= (C + µi − µj) coth
(
C + µi − µj
2kBΘ
)
.
C.3 The Bose-Einstein Distribution
Identical indistinguishable particles with integer spin obey the Bose-Einstein (BE)
statistics, as a governing principle in the quantum world. A example of BE distribu-
tion is the Planck radiation formula.
fBE(E, µ, T ) =1
e(E−µ)/kBT − 1
The BEredu to the MB distribution in the limit of high temperature and low
density.
C.4 Basic Distribution Functions
A. Binomial Distribution Function
It is the probability of an event occuring at x times out of n trials with a success
probability p in a single trial.
fb(x) =n!px(1 − p)n−x
x!(n− x)!
MEAN = np
STANDARD DEVIATION =√
np(1 − p)
B. Gaussian Distribution Function
The limit of the binomial distribution function at a large n is known as Gaussian
C.4. BASIC DISTRIBUTION FUNCTIONS 175
or normal distribution function. With a mean a and a standard deviation σ,
fg(x) =1√
2πσ2exp
[
−(x− a)2
2σ2
]
.
C. Poisson Distribution Function
Poisson distribution function is another limit of the binomial distribution func-
tion for a small probability p, expressed as a mean a
fp(x) =e−aax
x!
Appendix D
The Dirac Delta Function
D.1 Representations
•δ(x− x0) =
1
2π
∫ ∞
−∞dkeik(x−x0)
•δ(x) = lim
g→∞
sin(gx)
πx
•
δ(x) =1
2πlimg→∞
∫ g
−g
dkeikx
=1
2πlimg→∞
eigx − e−igx
ix
= limg→∞
sin(gx)
πx
•δ(x) = lim
a→0
1
π
a
x2 + a2
•δ(x) = lim
a→∞
a√πe−a2x2
176
D.2. PROPERTIES 177
D.2 Properties
•∫
dxδ(x− x0) = 1
•∫
dxδ(x)f(x) = f(0)
• δ∗(x) = δ(x)
• δ(−x) = δ(x)
• δ(ax) = 1|a|δ(x)
• f(x)δ(x− a) = f(a)δ(x− a)
• δ(x2 − a2) = 12|a| [δ(x− a) + δ(x+ a)]
•∫
δ(x− b)δ(a− x)dx = δ(a− b)
•∫
dxδ′(x)f(x) = −f ′(0)
• δ(x− a) = dΘ(x−a)dx
Appendix E
Useful Mathematical Formulas
E.1 Even and Odd functions
∫ l
−l
f(x)dx =
0 if f(x) is odd,
2∫ l
0f(x)dx if f(x) is even.
E.2 Taylor Series
Definition
f(x) = f(a) + f ′(a)(x− a) + f′′
(a)2!
(x− a)2 + ...fn(a)n!
(x− a)n + ...
Examples
• ex =∑∞
n=0xn
n!for all x
• ln(1 + x) =∑∞
n=0(−1)n
n+1xn+1 for |x| < 1
• xm
1−x=∑∞
n=m xn for |x| < 1
• (1 + x)α =∑∞
n=0
(
α
n
)
xn for all |x| < 1 and all α ∈ C
• sin x =∑∞
n=0(−1)n
(2n+1)!x2n+1 for all x
178
E.3. FOURIER-TRANSFORM 179
E.3 Fourier-Transform
Fourier transforms are commonly used in various fields. They are very useful to
convert information easily between two conjugate variables such as position x and
momentum p, energy E and time t or frequency ω and time t.
Continuous Fourier Transform
Suppose f(t) is square-integrable function. It can be decomposed by complex expo-
nentials with frequency components F(ω)
f(t) =1√2π
∫ ∞
−∞F(ω)eiωtdω.
The inverse Fourier transform is also defined as
F(ω) =1√2π
∫ ∞
−∞f(t)e−iωtdt.
The coefficient 1/√
2π is chosen symmetrically for both t and ω, which is known as
unitary. A common non-unitary transform is
f(t) =1
2π
∫ ∞
−∞F(ω)eiωtdω.
F(ω) =
∫ ∞
−∞f(t)e−iωtdt.
Finite Fourier Transform
Instead of continuous variable, finite Fourier transform is often employed handling
given number of data. It is called the discrete Fourier transform as well. Suppose N
complex numbers x0, ..., xN−1 which are converted into the sequency of N complex
numbers X0, ..., XN−1. The Fourier and inverse Fourier transforms are as follows:
Xk =N−1∑
n=0
xne− 2πi
Nkn k = 0, ...., N − 1,
180 APPENDIX E. USEFUL MATHEMATICAL FORMULAS
xn =1
N
N−1∑
k=0
Xke2πiN
kn n = 0, ..., N − 1
E.4 Pauli Spin Matrices
Representation
σx =
(
0 1
1 0
)
, σy =
(
0 −ii 0
)
, σz =
(
1 0
0 −1
)
.
Properties
• σi · σj = 2δij, for i 6= j.
• [σi, σj] = 2iǫijkσk.
• det(σi) = −1.
• Tr(σi) = 0.
• σ†i = σi.
• (~σ · ~a)(~σ ·~b) = ~a ·~b+ i~σ · (~a×~b).
• (~σ · ~a)2 = |~a|2.
E.5 Trigonometric Functions
A,B are the angle in radian.
• sin2A+ cos2A = 1.
• sec2A− tan2A = 1.
• sin(A±B) = sinA cosB ± cosA sinB.
• cos(A±B) = cosA cosB ∓ sinA sinB.
E.6. SPECIAL FUNCTIONS 181
• tan(A±B) = tan A±tan B1∓tan A tan B
.
• sinA± sinB = 2 sin (A±B)2
cos (A∓B)2
.
• cosA+ cosB = 2 cos (A+B)2
cos (A−B)2
.
• cosA− cosB = 2 sin (A+B)2
sin (B−A)2
.
• sin 2A = 2 sinA cosA = 2 tan A1+tan2 A
.
• cos 2A = cos2A− sin2A = 2 cos2A− 1 = 1 − 2 sin2A
= 1−tan2 A1+tan2 A
= cot A−tan Acot A+tan A
.
• tan 2A = 2 tan A1−tan2 A
= 2 cot Acot2 A−1
= 2cot A−tan A
.
• sin2A = 1−cos 2A2
.
• cos2A = 1+cos 2A2
.
E.6 Special Functions
• Gamma function
Γ(p) =
∫ ∞
0
xp−1e−xdx, p > 0.
• Error function
erf(x) =2√π
∫ x
0
e−t2dt.
• Lorentzian function
L(x) =2α
x2 + α2.
• Gaussian function
G(x) = ae−(x−b)2/c2 .
182 APPENDIX E. USEFUL MATHEMATICAL FORMULAS
E.7 Vector Operators
Some vector operators frequently used in various occasions are summarized in three
different coordinates: Cartesian, cylindrical and spherical. Note that the angle θ
in the spherical coordinate is defined in the x − y plane such as in the cylindrical
coordinates ad φ is defined the angle from the z-axis.
Gradient
A. Cartesian Coordinates
~∇ = i∂
∂x+ j
∂
∂y+ k
∂
∂z
B. Cylindrical Coordinates
~∇ = r∂
∂r+ θ
1
r
∂
∂θ+ z
∂
∂z
C. Spherical Coordinates
~∇ = r∂
∂r+ φ
1
r
∂
∂φ+ θ
1
r sinφ
∂
∂θ
Divergence
A. Cartesian Coordinates
~∇ · ~F =∂Fx
∂x+∂Fy
∂y+∂Fz
∂z
B. Cylindrical Coordinates
~∇ · ~F =1
r
∂(rFr)
∂r+
1
r
∂Fθ
∂θ+∂Fz
∂z
C. Spherical Coordinates
~∇ · ~F =1
r2
∂(r2Fr)
∂r+
1
r
∂Fθ
∂θ+
1
r sinφ
∂Fφ sinφ
∂φ
E.7. VECTOR OPERATORS 183
Curl
A. Cartesian Coordinates
~∇× ~F =
∣
∣
∣
∣
∣
∣
∣
∣
i j k∂∂x
∂∂y
∂∂z
Fx Fy Fz
∣
∣
∣
∣
∣
∣
∣
∣
B. Cylindrical Coordinates
~∇× ~F =1
r
∣
∣
∣
∣
∣
∣
∣
∣
r θ z∂∂r
∂∂θ
∂∂z
Fr rFθ Fz
∣
∣
∣
∣
∣
∣
∣
∣
C. Spherical Coordinates
~∇× ~F =
∣
∣
∣
∣
∣
∣
∣
∣
1r2 sin φ
r 1r sin φ
φ 1rθ
∂∂r
∂∂φ
∂∂θ
Fr rFφ r sinφFθ
∣
∣
∣
∣
∣
∣
∣
∣
Laplacian
A. Cartesian Coordinates
∇2 =∂2
∂x2+
∂2
∂y2+
∂2
∂z2
B. Cylindrical Coordinates
∇2 =1
r
∂
∂r
(
r∂
∂r
)
+1
r2
∂2
∂θ2+
∂2
∂z2
C. Spherical Coordinates
∇2 =1
r2
∂
∂r
(
r2 ∂
∂r
)
+1
r2 sin2 φ
∂2
∂θ2+
1
r2 sinφ
∂
∂φ
(
sinφ∂
∂φ
)
Appendix F
Recipe of Making Printed Circuit
Boards
This appendix introduces a simple recipe to build a prototype of printed circuit
boards (PCB) with a resolution around 10 µm in a quick and dirty method. The
PCBs generated by this recipe works quite well using photolithography and etching.
Step 1: Photomask Design
1. Generate a pattern of PCBs using softwares such as power point, illustrator or
CAD.
2. Print the pattern in the overhead projector transparencies with a laser printer.
3. Make sure all structures and patterns come out as black to block the light. If not,
paint thoroughly the incomplete structures and/or patterns with black marker pens.
Step 2: Substrate Preparation
1. Clean the surface of substrates. For Cu PCBs, a sand paper works well.
2. Resolve the grease by soaking substrates into the following sequential solvents:
Acetone → Methanol → Isopronanol
3. If critical, ultrasonic the substrates in Leksol, the detergent about 20 minutes.
184
185
Step 3: Resist Coating
1. Spin one side of the substrate with Shipley 1813 for 40 - 60 seconds at 2000 -
3000 rpm. Typically, 2000 rpm gives 2 µm-thick resist coatig and 3000 rpm coats 1
µm-thick resist on the top. (NOTE: Any thickness between 1 and 2 µm works well.)
2. Bake the coated substrate for 20 minutes in the oven at 80 oC in order to remove
remaining solvents in the resist.
3. Do the other side of the substrate, following the above for protection from etching
if necessary.
Step 4: Exposure
In the case of the EV aligner in Ginzton Clean room,
1. Turn on the lamp for more than 10 minutes before actual exposure. (NOTE: Do
not push the button too long, around 10 seconds are enough to enlighten the lamp.)
2. Power on the EV aligner.
3. Press “Load” button to place the substrate and mask.
4. Flip the “Shutter open” to expose the light for 48 seconds.
5. Flip the “Shutter open” back to the original position to stop exposure.
Step 5: Develop
1. Prepare the developing solution by mixing the Develop and water as a 1:1 ratio.
2. Develop the substrate around 45 seconds. (NOTE: around 10 seconds, the pattern
starts to appear.)
3. Clean the substrate in water.
Step 6: Etching
1. Prepare two beakers: One is for etchant and the other for clean water.
2. Soak the substrate into the etchants.
(a) Cu substrate: Ferric Chloride for more than 40 minutes. (NOTE: Heating up
the substrate may facilitate the etching process.)
186 APPENDIX F. RECIPE OF MAKING PRINTED CIRCUIT BOARDS
(b) Au (Cr) substrate: Au (Cr) etchant for a few minutes
3. Alternate two beakers of the etchant and water for speeding up the etching.
Step 7: Cleaning
1. Clean the substrate in the running distilled water.
2. Blow up the substrate with a nitrogen gun.
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